################################################################################ # GenBeukersZeta3.txt Save this file as GenBeukersZeta3.txt to use it, # # stay in the # ## same directory, get into Maple (by typing: maple ) # ## and then type: read GenBeukersZeta3.txt # ## Then follow the instructions given there # ## # ## Written by Robert Dougherty-Bliss, Christoph Koutschan and Doron Zeilberger # ################################################################################ print(` Written: Sept.-Nov. 2020 `): print(): print(`This is GenBeukersZeta3.txt, A Maple package for efficient computation of generalized Beukers integrals`): print(`that generalize Zeta(3) `): print(` It also experiments with the Generalized Beukers integral`): print(Int(Int(Int(x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(c1)*(1-z)^(c2)/(1-z+x*y*z)^(d+1)*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n,x=0..1),y=0..1),z=0..1)): print(``): print(` It is one of the packages that accompanies the article `): print(`Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs A La Apery`): print(``): print(`by Robert Dougherty-Bliss, Christoph Koutschan and Doron Zeilberger `): print(``): print(`avaliable from the authors' websites and from arxiv.org `): print(``): print(): print(`The most current version is available on WWW at:`): print(` http://sites.math.rutgers.edu/~zeilberg/tokhniot/GenBeukersZeta3.txt .`): print(`Please report all bugs to: DoronZeil at gmail dot com .`): print(): print(`---------------------------------------------`): print(): print(`For a list of the STORY procedures`): print(` type "ezraST();". For specific help type "ezra(procedure_name);" `): print(): print(): print(`---------------------------------------------`): print(): print(): print(`---------------------------------------------`): print(): print(`For a list of the Pre-Computed procedures`): print(` type "ezraPC();". For specific help type "ezra(procedure_name);" `): print(): print(): print(`---------------------------------------------`): print(): print(): print(`---------------------------------------------`): print(): print(`For general help, and a list of the MAIN functions,`): print(` type "ezra();". For specific help type "ezra(procedure_name);" `): #print(`For a list of the supporting functions type: ezra1();`): print(): print(): print(`---------------------------------------------`): print(): ezraPC:=proc() if args=NULL then print(` The Pre-computed procedures are`): print(` Hopefuls1, Hopefuls1C, Hopefuls2, Z3const `): else ezra(args): fi: end: ezraST:=proc() if args=NULL then print(` The Story procedures are`): print(` LongBook1, LongBook2, PrintPpG, ShortBook1,ShortBook2, TheoremZ3 `): else ezra(args): fi: end: ezra1:=proc() if args=NULL then print(`The supporting procdedures are`): print(` AsyPp, AsyPpG, CKtoB, EvalPp, EvalPpG, ExtractPower, ExtractPp, Fs, GBCZ31ck, Guess1,`): print(`GuessPp1, Kefel, Khalek, Lc, MyID, MyIDs, NorOp, Pp, PpG, PpLimit, PpGlimit, PrimesF, RoundRat, Seg1, SortC `): else ezra(args): fi: end: ezra:=proc() if args=NULL then print(`The MAIN procedures are:`) : print(` An,Bn, AnBn, AppxSeq1, CnDn, Cn,Dn, delt, deltSeqZ3, FindRelGZ3, GBCZ3ck, Guess, OPEZ3, Info1, IntGBZ3, IntGBZ3ck, Search, SearchNew `): elif nargs=1 and args[1]=An then print(`An(a,b,c,d,e,n): The numerator, An, in the `): print(`for IntGBZ3(b, c, e, a, a, c, d,n):`): print(`Try: `): print(` An(0,0,0,0,0,50); `): elif nargs=1 and args[1]=AnBn then print(`AnBn(a,b,c,d,e,K): The first K terms in the sequences An and Bn in the expression`): print(`for IntGBZ3(b, c, e, a, a, c, d,n):`): print(`Try: `): print(` AnBn(0,0,0,0,0,50); `): elif nargs=1 and args[1]=AppxSeq1 then print(`AppxSeq1(ope,n,N,RE,K): It inputs a second order operator ope, in n where N is the shift operator N`): print(`where it is known that ope(n,N) annihilates a sequence a(n) that goes to zero.`): print(`It also inputs a pair RE=[[c0,c1],L] satisfying c0*a(0)+c1*a(1)=L where c0,c1 are integers and`): print(`L is a rational number, It outputs the list of the first K terms in the rational approximations to a(0). Try`): print(`AppxSeq1(N^2-N-1,n,N,[[11,12],3],20);`): elif nargs=1 and args[1]=AsyPp then print(`AsyPp(P): inuts a Pp expression of the form outputted by Guess, and outputs the`): print(`EXACT value of limit of log( EvalPp(P,n))/n as n goes to infinity. Try`): print(`AsyPp([[3, 3], [[3, 1], [[3, [1/3, 2/3]], [3/2, [2/3, 1]]]]]);`): elif nargs=1 and args[1]=AsyPpG then print(`AsyPpG(P): inputs a Pp expression of the form outputted by Guess, and outputs the`): print(`EXACT value of limit of log( EvalPpG(n))/n as n goes to infinity. Try: `): print(`For GBCZ3ck(0,0,0,0,0) (alias Zeta(3)), type:`): print(` AsyPpG([[1, 1], [1, 3],[] ]);`): print(``): print(`For GBCZ3ck(0,1/2,1/2,0,1/2) ( that equals (conjecturally) -198 + 288 ln(2)), type:`): print(`AsyPpG([[1, 1], [2, 1],[] ] );`): print(`For GBCZ3ck(0,0,0,1/2,1/2) ( conjecturally -(2-4*ln(2))/(3-4*ln(2))), type:`): print(`AsyPpG([[2, 4], [1, 0],[ [0,1/2,0,1,{0},1], [1/2,1,0,2,{0},1 ]$3 ]] );`): print(`For GBCZ3ck(0,1/4,0,3/4,1/2), conjecturally -(-1+ln(2))/(-2+3*ln(2)), type:`): print(`AsyPpG([[2, 6], [4, 1], [ [1/4,3/4,0,2,{0},1], [3/4,1,0,4/3,{0},1], [3/4,1,0,4/3,{0},1] ] ] );`): print(``): print(`For GBCZ3ck(0,0,0,1/3,2/3), type`): print(`AsyPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1], [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ]);`): print(``): print(`For GBCZ3ck(0,0,0,2/3,1/3), type`): print(`AsyPpG([[3, 3], [3, 1], [[2/3, 1, 0, 3/2, {0}, 1]$2, [1/3, 2/3, 0, 3/4, {0}, 1], [1/3, 2/3, 3/2, 3, {1}, 3]]]);`): print(``): print(`For GBCZ3ck(0, 1/3, 2/3, 1/3, 2/3), type`): print(`AsyPpG([[3, 3/2], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$3, [2/3, 1, 0, 3/2, {0}, 1]$2, [0, 1/3, 0, 1, {0}, 1]]]);`): print(``): print(`For GBCZ3ck(0, 1/5, 0, 3/5, 2/5), type`): print(`AsyPpG([[5, 5/2], [5, 1], [[4/5, 1, 0, 5/4, {0}, 1]$2, [2/5, 4/5, 0, 5/2, {0}, 1]]]);`): #print(``): #print(`For The equivalent GBCZ3ck(4/5, 2/5, 1/5, 3/5, 0), BUT it is STRICLY CONTAINS IT, SO IT HAS TO BE ADJUSTED. Type"`): #print(`AsyPpG([[1, 1], [1, 0], [[0, 1, 0, 1, {0}, 1],[1/2, 1, 1, 2, {0}, 1],[1/3, 1/2, 2, 3, {0}, 1],[1/4, 1/3, 3, 4, {0}, 1],[1/5, 1/4, 4, 5, {0}, 1] ]]);`): #print(``): #print(`For GBCZ3ck(0, 1/2, 1/2, 1/3, 1/6), type`): #print(`AsyPpG([[3, 3], [5, 1], [[4/5, 1, 0, 5/4, {0}, 1]$2, [2/5, 4/5, 0, 5/2, {0}, 1]]]);`): elif nargs=1 and args[1]=Bn then print(`Bn(a,b,c,d,e,n): The denominator Bn, in the `): print(`for IntGBZ3(b, c, e, a, a, c, d,n):`): print(`Try: `): print(` Bn(0,0,0,0,0,50); `): elif nargs=1 and args[1]=CnDn then print(`CnDn(a,b,c,d,e,K): The first K terms in the sequences gcd(numer(An[i]),number(Bn[i]))`): print(`lcm(denom(An[i]),denom(Bn[i])) where An, Bn are the sequences given by AnBn(a,b,c,d,e,K) (q.v.)`): print(`It also returns the last 20 terms of the logs of the normailizing sequence`): print(`as well as the last 20 terms of the implied deltas`): print(` Try: `): print(`CnDn(0,0,0,0,0,50);`): elif nargs=1 and args[1]=Cn then print(`Cn(a,b,c,d,e,n): The denominator, Cn, of the n-th term in the integerating sequence for `): print(`for IntGBZ3(b, c, e, a, a, c, d,n):`): print(`Try: `): print(` Cn(0,0,0,0,0,50); `): elif nargs=1 and args[1]=CKtoB then print(`CKtoB(a,b,c,d,e): :[b, c, e, a, a, c, d] converting from the Koutschan operator to the usual convention`): elif nargs=1 and args[1]=delt then print(` delt(a,c): Given a rational number a and a constant `): print(` c, finds the delta such that |a-c|=1/denom(a)^(1+delta) `): print(` For example, try delt(22/7,evalf(Pi)); `): elif nargs=1 and args[1]=deltSeqZ3 then print(`deltSeqZ3(a,b,c,d,e,K): the sequence of empirial deltas for the sequence approximating`): print(`It also returns [b, c, e, a, a, c, d]`): print(`IntGBZ3ck(b, c, e, a, a, c, d,0);`): print(`K must be at least 10. Try:`): print(`deltSeqZ3(0,0,0,0,0,30);`): elif nargs=1 and args[1]=Dn then print(`Dn(a,b,c,d,e,n): The numerator, Dn, of the n-th term in the integerating sequence for `): print(`for IntGBZ3(b, c, e, a, a, c, d,n):`): print(`Try: `): print(` Dn(0,0,0,0,0,50); `): elif nargs=1 and args[1]=EvalPp then print(`EvalPp(P,n1): evaluate a Pp object at n=n1. Try:`): print(`EvalPp([[3, 3], [[3, 1], [[29/10, [1/3, 2/3]], [3/2, [2/3, 1]]]]],1000);`): elif nargs=1 and args[1]=EvalPpG then print(`EvalPpG(P,n1): evaluate a PpG object at n=n1. `): print(`For GBCZ3ck(0,0,0,0,0) (alias Zeta(3)), type:`): print(` EvalPpG([[1, 1], [1, 3],[] ],1000);`): print(``): print(`For GBCZ3ck(0,1/2,1/2,0,1/2) ( that equals (conjecturally) -198 + 288 ln(2)), type:`): print(`EvalPpG([[1, 1], [2, 1],[] ] ,1000);`): print(`For GBCZ3ck(0,0,0,1/2,1/2) ( conjecturally -(2-4*ln(2))/(3-4*ln(2))), type:`): print(`EvalPpG([[2, 4], [1, 0],[ [0,1/2,0,1,{0},1], [1/2,1,0,2,{0},1 ]$3 ]] ,1000);`): print(`For GBCZ3ck(0,1/4,0,3/4,1/2), conjecturally -(-1+ln(2))/(-2+3*ln(2)), type:`): print(`EvalPpG([[2, 6], [4, 1], [ [1/4,3/4,0,2,{0},1], [3/4,1,0,4/3,{0},1], [3/4,1,0,4/3,{0},1] ] ] ,1000);`): print(``): print(`For GBCZ3ck(0,0,0,1/3,2/3), type`): print(`EvalPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1], [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ],1000);`): print(``): print(`For GBCZ3ck(0,0,0,2/3,1/3), type`): print(`EvalPpG([[3, 3], [3, 1], [[2/3, 1, 0, 3/2, {0}, 1]$2, [1/3, 2/3, 0, 3/4, {0}, 1], [1/3, 2/3, 3/2, 3, {1}, 3]]],1000);`): print(``): print(`For GBCZ3ck(0, 1/3, 2/3, 1/3, 2/3), type`): print(`EvalPpG([[3, 3/2], [1, 0], [[1/3, 2/3, 0, 3, {1}, 3]$3, [2/3, 1, 0, 3/2, {0}, 1]$2, [0, 1/3, 0, 1, {0}, 1]]],1000);`): print(``): print(`For GBCZ3ck(0, 1/5, 0, 3/5, 2/5), type`): print(`EvalPpG([[5, 5/2], [5, 1], [[4/5, 1, 0, 5/4, {0}, 1]$2, [2/5, 4/5, 0, 5/2, {0}, 1]]],1000);`): #print(``): #Bprint(`For The equivalent GBCZ3ck(4/5, 2/5, 1/5, 3/5, 0), BUT it is STRICLY CONTAINS IT, SO IT HAS TO BE ADJUSTED. Type"`): #print(`EvalPpG([[1, 1], [1, 0], [[0, 1, 0, 1, {0}, 1],[1/2, 1, 1, 2, {0}, 1],[1/3, 1/2, 2, 3, {0}, 1],[1/4, 1/3, 3, 4, {0}, 1],[1/5, 1/4, 4, 5, {0}, 1] ]],1000);#`): #print(``): #print(`For GBCZ3ck(0, 1/2, 1/2, 1/3, 1/6), type`): #print(`EvalPpG([[3, 3], [5, 1], [[4/5, 1, 0, 5/4, {0}, 1]$2, [2/5, 4/5, 0, 5/2, {0}, 1]]],1000);`): elif nargs=1 and args[1]=ExtractPower then print(`ExtractPower(N,n,K,eps): inputs a positive integer N decides whether there is a factor close to c^(a*n) for some constant c `): print(` and a simple`): print(`rational number with denominator less than K. Try: `): print(` ExtractPower(7^1497,1000,4,0.01); `): print(` ExtractPower(CnDn(0,3/5,4/5,2/5,-4/5,2000)[1][1][-1],2000,5,0.01); `): print(` ExtractPower(CnDn(4/5,3/5,0,3/5,1/5,2000)[1][1][-1],2000,5,0.01); `): print(` ExtractPower(CnDn(0,1/4,0,3/4,1/2,4000)[1][1][-1],4000,20,0.01);`): elif nargs=1 and args[1]=ExtractPp then print(`ExtractPp(N,n,K1,K2,r,eps): inputs a large integer N belonging to some much smalller integer n (N is the n-th member of some sequence, in real life)`): print(`Conjectures an approrixmation to it in the form lcm(1..A*n)*Pp(a1,b1,n,n*c1)^1*Pp(a2,b2,n,n*c2)^2*...`): print(`The output is [A,[a1,b1,c1],[a2,b2,c2], ...]. K1,K2,r, eps are guessing parameters. For example, try:`): print(`ExtractPp(CnDn(0,1/4,0,3/4,1/2,1000)[1][2][-1],1000,10,5,5,0.05);`): print(`ExtractPp(CnDn(0,1/4,0,3/4,1/2,2000)[1][2][-1],2000,10,5,5,0.05);`): print(`ExtractPp(CnDn(0,1/4,0,3/4,1/2,3000)[1][2][-1],3000,10,5,5,0.05);`): elif nargs=1 and args[1]=FindRelGZ3 then print(`FindRelGZ3(a1,a2,b1,b2,c1,c2,d): A clever way to find a relationship`): print(`between IntGBgZ3(a1,a2,b1,b2,c1,c2,d,0) and IntGBgZ3(a1,a2,b1,b2,c1,c2,d,1) . Try:`): print(`FindRelGZ3(-1/2,0,0,1/2,0,0,0);`): elif nargs=1 and args[1]=Fs then print(`Fs(S,n): Given a set S and a positive integer n finds sort([seq(evalf(frac(n/p),10),p in S)]);`): elif nargs=1 and args[1]=GBCZ31ck then print(`GBCZ31ck(a,b,c,d,e, K): A floating-point approximation to the constant`): print(`IntGBZ3(b, c, e, a, a, c, d) supposed to be good for the number of digits`): print(`using the sequence to K terms `): print(`followed by the differene of two consecutive terms. Try:`): print(`GBCZ31ck(0,0,0,0,0,100);`): elif nargs=1 and args[1]=Guess then print(`Guess(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC, CUTOFF): Guesses an "Integrating Factor" for the irrationality of GBCZ3ck(a,b,c,d,e) (q.v.)`): print(`with guessing parameters K1,K2,r,eps. and the max CUTOFF of the deviation. It inestigates 1000*i from i=1 to M.`): print(`P(a,b,m,n) stands for Pp(a,b,m,n) and LC(a) for lcm(1..a) `): print(`It returns the approximation to Dn followed by the approximation of Cn`): print(``): print(``): print(`Try:`): print(`Guess(0,1/4,0,3/4,1/2,10,10,6,0.05,2,n,P,LC,10^50); `): print(` For Zeta(3) `): print(`Guess(0,0,0,0,0,10,10,6,0.05,2,n,P,LC,10^50); `): print(`Also try:`): print(``): print(`For the first constant with parameters with denominator 3`): print(`Guess( 0, 0, 0, 1/3, 2/3,5,5,6,0.1,3,n,P,LC,10^50);`): print(``): print(`For the second constant with parameters with denominator 3`): print(`Guess( 0, 0, 0, 2/3, 1/3,5,5,6,0.1,3,n,P,LC,10^10);`): print(``): print(`For the third constant with parameters with denominator 3`): print(`Guess( 0, 1/3, 2/3, 1/3, 2/3,5,5,6,0.1,3,n,P,LC,10^10);`): print(``): print(`For the only constant with parameters with denominator 5`): print(`Guess(0,1/5,0,3/5,2/5,10,10,6,0.05,3,n,P,LC,10^10); `): print(``): print(`For the first constant with parameters with denominator 6`): print(`Guess(0, 1/2, 1/2, 1/3, 1/6,10,10,6,0.1,1,n,P,LC,10^10); `): print(``): print(`For the second constant with parameters with denominator 6`): print(`Guess(0, 1/2, 1/2, 1/6, 1/3,10,10,6,0.1,3,n,P,LC,10^10); `): print(``): print(`For the third constant with parameters with denominator 6`): print(`Guess(1/3, 0, 2/3, 1/2, 5/6,10,10,6,0.05,3,n,P,LC,10^10);`): print(``): print(`For the first constant with parameters with denominator 7 (so far it gives FAIL)`): print(`Guess(1/7, 0, 2/7, 3/7, 4/7,10,10,6,0.1,1,n,P,LC,10^10); `): print(``): print(`For the second constant with parameters with denominator 7`): print(`Guess(1/7, 0, 2/7, 5/7, 3/7,10,10,6,0.1,2,n,P,LC,10^10); `): print(``): print(`For the third constant with parameters with denominator 7 (so far it gives FAIL) `): print(`Guess( 1/7, 0, 3/7, 4/7, 5/7,10,10,6,0.1,2,n,P,LC,10^10); `): print(``): print(`For the fourth constant with parameters with denominator 7 `): print(`Guess( 1/7, 0, 4/7, 2/7, 5/7,10,10,6,0.1,1,n,P,LC,10^10); `): print(``): print(`For the fifth constant with parameters with denominator 7 (so far it gives FAIL) `): print(`Guess( 2/7, 0, 3/7, 4/7, 5/7,10,10,6,0.1,1,n,P,LC,10^10); `): elif nargs=1 and args[1]=Guess1 then print(`Guess1(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC): Guesses an "Integrating Factor" for the irrationality of GBCZ3ck(a,b,c,d,e) (q.v.)`): print(`with guessing parameters K1,K2,r,eps. It inestigates 1000*i from i=1 to M.`): print(`P(a,b,m,n) stands for Pp(a,b,m,n) and LC(a) for lcm(1..a) `): print(`It returns the approximation to Dn followed by the approximation of Cn`): print(`It also returns the ratio of the guess to the real thing `): print(``): print(`Try:`): print(`Guess1(0,1/4,0,3/4,1/2,10,10,6,0.05,2,n,P,LC); `): print(` For Zeta(3) `): print(`Guess1(0,0,0,0,0,10,10,6,0.05,2,n,P,LC); `): print(`Also try:`): print(``): print(`For the first constant with parameters with denominator 3`): print(`Guess1( 0, 0, 0, 1/3, 2/3,5,5,6,0.1,3,n,P,LC);`): print(``): print(`For the second constant with parameters with denominator 3`): print(`Guess1( 0, 0, 0, 2/3, 1/3,5,5,6,0.1,3,n,P,LC);`): print(``): print(`For the third constant with parameters with denominator 3`): print(`Guess1( 0, 1/3, 2/3, 1/3, 2/3,5,5,6,0.1,3,n,P,LC);`): print(``): print(`For the only constant with parameters with denominator 5`): print(`Guess1(0,1/5,0,3/5,2/5,10,10,6,0.05,3,n,P,LC); `): print(``): print(`For the first constant with parameters with denominator 6`): print(`Guess1(0, 1/2, 1/2, 1/3, 1/6,10,10,6,0.1,1,n,P,LC); `): print(``): print(`For the second constant with parameters with denominator 6`): print(`Guess1(0, 1/2, 1/2, 1/6, 1/3,10,10,6,0.1,3,n,P,LC); `): print(``): print(`For the third constant with parameters with denominator 6`): print(`Guess1(1/3, 0, 2/3, 1/2, 5/6,10,10,6,0.05,3,n,P,LC);`): print(``): print(`For the first constant with parameters with denominator 7 (so far it gives FAIL)`): print(`Guess1(1/7, 0, 2/7, 3/7, 4/7,10,10,6,0.1,1,n,P,LC); `): print(``): print(`For the second constant with parameters with denominator 7`): print(`Guess1(1/7, 0, 2/7, 5/7, 3/7,10,10,6,0.1,2,n,P,LC); `): print(``): print(`For the third constant with parameters with denominator 7 (so far it gives FAIL) `): print(`Guess1( 1/7, 0, 3/7, 4/7, 5/7,10,10,6,0.1,2,n,P,LC); `): print(``): print(`For the fourth constant with parameters with denominator 7 `): print(`Guess1( 1/7, 0, 4/7, 2/7, 5/7,10,10,6,0.1,1,n,P,LC); `): print(``): print(`For the fifth constant with parameters with denominator 7 (so far it gives FAIL) `): print(`Guess1( 2/7, 0, 3/7, 4/7, 5/7,10,10,6,0.1,1,n,P,LC); `): elif nargs=1 and args[1]=Hopefuls1 then print(`Hopefuls1(): A list of length 7 where the entries are lists of hopepuls quintuples [a,b,c,d,e], with denominators 1,2,3,4,5,6,7 respectively.`): print(`where a,b,c,d,e are positive`): print(`It is the output of CnDn(a,b,c,d,e,400)[3] are all positive. In other words, it is the pre-computed output of`): print(`Search(N,K) for N=1..6 `): print(`Note that the first 5 members are included in Hopefuls2()`): print(`Try: `): print(` Hopefuls1(); `): elif nargs=1 and args[1]=Hopefuls1C then print(`Hopefuls1C(): The pre-computed list of length 7 whose i-th entry are all the`): print(`hopefuls five-tuples such that the i-th entry is `): print(`the list of equivalence classes under linear-fractional transformations`): print(`gotten from Hopefuls1. The 2nd and 4th entry are equivalent to log(2). `): print(`It calls the GBCZ3ck of the j-th five-tuples in each list r[j].`): print(`Try:`): print(`Hopefuls1C(r);`): elif nargs=1 and args[1]=Hopefuls2 then print(`Hopefuls2(): A list of length 5 where the entries are lists of hopepuls quintuples [a,b,c,d,e], with denominators 1,2,3,4,5, respectively.`): print(`where a,b,c,d,e are negative or positive`): print(`It is the output of CnDn(a,b,c,d,e,400)[3] are all positive. In other words, it is the pre-computed output of`): print(`SearchNew(N,K) for N=1..5 `): print(`Try: `): print(` Hopefuls2(); `): elif nargs=1 and args[1]=Info1 then print(`Info1(a,b,c,d,e,K): inputs a five-tuple (a,b,c,d,e) and outputs a list consisting of`): print(`(i)the parameters [a,b,c,d,e]`): print(`(ii) the approximate value of the constant`): print(`(iii) the recurrence operator annihilating An and Bn (iv) the pair of initial conditions for An and Bn`): print(`i.e. [[A(0),A(1)],[B(0),B(1)]] (v) the estimated value of the limit of the log of the INTEGERATING functor divided by n (vi) the`): print(`estimated implied irrationality measure using K values (vii) an identification if successful. Try:`): print(`Info1(0,0,0,0,0,2000);`): elif nargs=1 and args[1]=IntGBZ3 then print(` IntGBZ3(a1,a2,b1,b2,c1,c2,d,n): `): print(`int(int(int(x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(c1)*(1-z)^(c2)/(1-z+x*y*z)^(d+1)*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n,z=0..1),x=0..1),y=0..1)):`): print(`Try: `): print(`IntGBZ3(0,0,0,0,0,0,0,0);`): elif nargs=1 and args[1]=IntGBZ3ck then print(` IntGBZ3ck(a,b,c,d,e,n): Same as IntGBZ3(b, c, e, a, a, c, d,n)`): print(`IntGBZ3ck(0,0,0,0,0,0);`): elif nargs=1 and args[1]=GBCZ3ck then print(`GBCZ3ck(a,b,c,d,e): A floating-point approximation to the constant`): print(`IntGBZ3(b, c, e, a, a, c, d) supposed to be good for the number of digits`): print(`It also returns [b, c, e, a, a, c, d]`): print(` Try: `): print(` GBCZ3ck(0,0,0,0,0); `): elif nargs=1 and args[1]=GuessPp1 then print(`GuessPp1(S,n,K2,eps): inputs a set of primes S and a positive integer n, and guessing parameters K2 (a positive intger) and eps `): print(`a small positive number`): print(`Conjectures rational numbers [a,b] such that frac(n/p) is between a and b. `): print(`It also returns the rational `): print(` number c such that conjecturally the largest prime in S is close to c*n . The output is `): print(` [c,[a,b]] or [c,[a]] . Try:`): print(`N1:= CnDn(0,1/4,0,3/4,1/2,2000)[1][2][-1]/Lc(8000); `): print(`GuesstPp1(PrimesF(N1,2000,5)[1][1],2000,5,0.01);`): elif nargs=1 and args[1]=Kefel then print(`Kefel(ope1,ope2,n,N): The product ope1*ope2 of two linear recurrence operators. Try: `): print(`Kefel(N-n-1,N-n^2,n,N); `): elif nargs=1 and args[1]=Khalek then print(`Khalek(Ope3,Ope2,n,N): Given a monic third-order operator Ope3 in n and the shift operator N`): print(`and a second-order one, finds the operators Ope1a Ope1b such that`): print(`Ope3=Ope1a*Ope2+Ope1b. If Ope3 is a left multiple of Ope2, Ope1b should be zero.`): print(`Try: `): print(` Khalek(N^3-1,N^2-1,n,N); `): elif nargs=1 and args[1]=Lc then print(`Lc(n): the lcm of 1, ..., n. Try: Lc(1000);`): elif nargs=1 and args[1]=LongBook1 then print(`LongBook1(K): A long book with statements of parameters that are very likely to produce irrationality proofs`): print(`followed by the conjectured irrationality measure`): print(`and COMPLETE sketch of proof (modulo a divisibility lemma left to the reader (and that we are unable to do yet))`): print(`K is a large positive integer that for estimating the conjectured delta.`): print(`It uses the database Hopefuls1(). Try: `): print(`LongBook1(300):`): elif nargs=1 and args[1]=LongBook2 then print(`LongBook2(K): A long book with statements of parameters that are very likely to produce irrationality proofs`): print(`followed by the conjectured irrationality measure`): print(`and COMPLETE sketch of proof (modulo a divisibility lemma left to the reader (and that we are unable to do yet))`): print(`K is a large positive integer that for estimating the conjectured delta.`): print(`It uses the database Hopefuls2(). Try: `): print(`LongBook2(300):`): elif nargs=1 and args[1]=MyID then print(`MyID(C,F,N): Given a constant C in floating point and a famous constant F (like log(2)) and a positive integer N`): print(`tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ`): print(`MyID(evalf((log(2)-2)/(2*log(2)+3)),log(2),100);`): elif nargs=1 and args[1]=MyIDs then print(`MyIDs(C,F,F0,N): Given a constant C in decimals and another constant`): print(`let's call it F, whose floating-point is F0`): print(` and a positive integer N`): print(`tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ`): print(`MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100);`): elif nargs=1 and args[1]=NorOp then print(`NorOp(ope,N): normalizes the operator ope. Try:`): print(`NoOp(n*N^2-N-1,N);`): elif nargs=1 and args[1]=OPEZ3 then print(`OPEZ3(a,b,c,d,e,n,Sn): The 2nd-order linear recurrence operator in n and the shift operator Sn annnihilating the`): print(`double integral over [0,1]x[0,1] of the function`): print(` x^a1*(1-x)^a2*y^b1*(1-y)^b2*z^c1*(1-z)^c2*(x*(1-x)*y*(1-y)*z*(1-z))^n/(1-z+x*y*z)^(n+d+1)`): print(` parameters: [a1,a2,b1,b2,c1,c2,d]=[b, c, e, a, a, c, d]`): print(`It also returns [b, c, e, a, a, c, d], so the output is a pair [ope, ListOfLength7]. Try:`): print(`OPEZ3(0,0,0,0,0,n,Sn);`): elif nargs=1 and args[1]=Pp then print(`Pp(a,b,n,m): Given rational numbers a and b between 0 and 1 and positive integers m and n`): print(`it is the product of all primes p less than m such that frac{n/p} is between a and b. Try`): print(`Pp(0,1/2,100,100);`): elif nargs=1 and args[1]=PpG then print(`PpG(a,b,n,m1,m2,C,M): Given rational numbers a and b between 0 and 1 and positive integers m1,m2 and n`): print(`and a subset C in {1,...M-1} and an integer M`): print(`it is the product of p between m1 and m2 such that frac{n/p} is between a and b`): print(`and p mod M is in C. Try`): print(`PpG(1/2,3/4,1000,500,1500,{1,3},5);`): elif nargs=1 and args[1]=PpLimit then print(`PpLimit(a, b, r): Return the limit of log(Pp(a, b, n, rn)) / n as n tends to infinity.`): print(`Try:`): print(`evalf([log(Pp(1/4, 3/4, 1000, 2 * 1000)) / 1000, PpLimit(1/4, 3/4, 2)]);`): print(`evalf([log(Pp(3/4, 1, 1000, 4 * 1000 / 3)) / 1000, PpLimit(3/4, 1, 4/3)]);`): print(`evalf([log(Pp(1/10, 1/2, 1000, 5 * 1000)) / 1000, PpLimit(1/10, 1/2, 5)]);`): elif nargs=1 and args[1]=PpGlimit then print(`The limit of log PpG(a,b,r1*n,r2*n,C,M)/n as n goes to infinity. Try`): print(`PpGlimit(1/3,2/3,3/2,3,{2},3);`): elif nargs=1 and args[1]=PrimesF then print(`PrimesF(N,n,k): inputs a HUGE integer N that is a product of small primes, and a much smaller integer n`): print(`that generated N (via some process), outputs the list of length k, whose i-th entry is the list`): print(`of primes>=sqrt(n) that show up with exponent i, followed by the list of primes that`): print(`did not show up at all. Try:`): print(`lu:=Hopefuls2()[3][1]: N:=CnDn(op(lu),2000)[1][2][-1]: PrimesF(N,2000,6); `): elif nargs=1 and args[1]=PrintPpG then print(`PrintPpG(P,n,L1,L2): P is the same input as the P EvalPpG(P,n1) (q.v.) but n,L1 and L2 are symbols`): print(`where n stands for n, L1(n) stands for lcm(1...n) and L2(alpha,beta,a,b,C,M,n) stands for the`): print(`product of all primes between a*n and b*n such that the fractional part of n/p is between alpha and beta`): print(`and p mod M belongs to the set C where C is a subset of the set of nonneg. integers less than M relatively prime to it.`): print(`If no modularity condition is present than M=1 and C={0}. Try:`): print( `PrintPpG([[1, 1], [ 1,3],[] ],n,LCM,P );`): print(`PrintPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1], [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ],n,LCM,P );`): print(`PrintPpG([[2, 6], [4, 1], [ [1/4,3/4,0,2,{0},1], [3/4,1,0,4/3,{0},1], [3/4,1,0,4/3,{0},1] ] ],n, LCM,P );`): elif nargs=1 and args[1]=rf then print(`rf(a,n): The raising factorial a(a+1)...*(a+n-1). Try:`): print(`rf(1/3,10);`): elif nargs=1 and args[1]=RoundRat then print(`RoundRat(ka,K,eps): inputs a number and a small postitive integer K, finds a rational number a/b with`): print(`denominator less than K such that abs(ka-a/b)nops(ini) then ERROR(`The length of`,ini, `should be `, degree(ope,N)): fi: gu:=ini: for i from ORD to L do lu:=0: n0:=i-ORD: lu:=0: for j from 0 to ORD-1 do lu:=lu+subs(n=n0,coeff(ope,N,j))*gu[n0+j+1]: od: lu:=-lu/subs(n=n0,coeff(ope,N,ORD)): gu:=[op(gu),lu]: od: gu: end: BCZ3:=proc():17+12*sqrt(2):end: #delt(a,c): Given a rational number a and a constant #c, finds the delta such that |a-c|=1/denom(a)^(1+delta) #For example, try delta(22/7,evalf(Pi)): delt:=proc(a,c): evalf(-log(abs(c-a))/log(denom(a)))-1: end: #AppxSeq1(ope,n,N,RE,K): It inputs a second order operator ope, in n where N is the shift operator N #where it is known that ope(n,N) annihilates a sequence a(n) that goes to zero. #It also inputs a pair RE=[[c0,c1],L] satisfying c0*a(0)+c1*a(1)=L where c0,c1 are integers and #L is a rational number, It outputs the list of the first K terms in the rational approximations to a(0). Try #AppxSeq1(N^2-N-1,n,N,[[11,12],3],20); AppxSeq1:=proc(ope,n,N,RE,K) local c0,c1,L,gu,a0,lu,i,mu0,mu1: if not (type(RE,list) and nops(RE)=2 and type(RE[1],list) and nops(RE[1])=2) then print(`Bad input`): RETURN(FAIL): fi: c0:=RE[1][1]: c1:=RE[1][2]: if c1=0 then RETURN(FAIL): fi: L:=RE[2]: gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K): gu:=[op(3..nops(gu),gu)]: lu:=[]: for i from 2 to nops(gu) do mu0:=coeff(gu[i],a0,0): mu1:=coeff(gu[i],a0,1): if mu1=0 then RETURN(FAIL): fi: lu:=[op(lu),-mu0/mu1]: od: lu: end: #FindRelGZ3(a1,a2,b1,b2,c1,c2,d): A clever way to find a relationship #between IntGBgZ3(a1,a2,b1,b2,c1,c2,d,0) and IntGBgZ3(a1,a2,b1,b2,c1,c2,d,1) . Try: #FindRelGZ3(1/2,0,0,1/2,0,0,0); FindRelGZ3:=proc(a1,a2,b1,b2,c1,c2,d) local x,y, c000,c100,c010,c001,c200,c020,c002,c110,c101,c011, d000,d100,d010,d001,d200,d020,d002,d110,d101,d011, e000,e100,e010,e001,e200,e020,e002,e110,e101,e011,gu,eq,var,lu,A,B,C,gu1,gu2,hal,ka,i,z,ka1,ka2,ka3,i1,i2,i3: option remember: lu:=x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(c1)*(1-z)^(c2)/(1-z+x*y*z)^d: gu1:=normal( diff((c000+c100*x+c010*y+c001*z+c200*x^2+c020*y^2+c002*z^2+c110*x*y+ c101*x*z+ c011*y*z)*x*(1-x)*lu/(1-z+x*y*z),x) +diff((d000+d100*x+d010*y+d001*z+d200*x^2+d020*y^2+d002*z^2+d110*x*y+ d101*x*z+ d011*y*z)*y*(1-y)*lu/(1-z+x*y*z),y) +diff((e000+d100*x+e010*y+e001*z+e200*x^2+e020*y^2+e002*z^2+e110*x*y+ e101*x*z+ e011*y*z)*z*(1-z)*lu/(1-z+x*y*z),z) ): gu1:=normal(gu1/lu): gu2:=normal(A/(1-z+x*y*z)+B*x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z)^2-C): gu:=normal(gu1-gu2): gu:=numer(gu): var:={c000,c100,c010,c001,c200,c020,c002,c110,c101,c011, d000,d100,d010,d001,d200,d020,d002,d110,d101,d011, e000,e100,e010,e001,e200,e020,e002,e110,e101,e011,A,B,C}: eq:={}: for i1 from 0 to degree(gu,x) do ka1:=coeff(gu,x,i1): for i2 from 0 to degree(gu,y) do ka2:=coeff(ka1,y,i2): for i3 from 0 to degree(gu,z) do ka3:=coeff(ka2,z,i3): eq:=eq union {ka3}: od: od: od: eq:=eq minus {0}: var:=solve(eq,var): hal:=subs(var,[A,B,C]): if hal[1]=0 then RETURN(FAIL): fi: hal:=[1,normal(hal[2]/hal[1]),normal(hal[3]/hal[1])]: hal: ka:=lcm(seq(denom(hal[i]),i=1..nops(hal))): hal:=ka*hal: if not (type(hal[1],numeric) and type(hal[2],numeric) and type(hal[3],numeric) ) then RETURN(FAIL): fi: [[hal[1],hal[2]],hal[3]]: end: #GBCZ31ck(a,b,c,d,e, K): A floating-point approximation to the constant #IntGBZ3(b, c, e, a, a, c, d) supposed to be good for the number of digits #using the sequence to K terms #followed by the differene of two consecutive terms. Try: #GBCZ31ck(0,0,0,0,0,100); GBCZ31ck:=proc(a,b,c,d,e,K) local RE, n,N,ope,lu: RE:=FindRelGZ3(b, c, e, a, a, c, d): if RE=FAIL then RETURN(FAIL): fi: ope:=OPEZ3(a,b,c,d,e,n,N): ope:=ope[1]: lu:=AppxSeq1(ope,n,N,RE,K): if lu=FAIL then RETURN(FAIL): fi: [evalf(lu[nops(lu)]),evalf(abs(lu[nops(lu)]-lu[nops(lu)-1]))]: end: #CKtoB(a,b,c,d,e): :[b, c, e, a, a, c, d] converting from the Koutschan operator to the usual convention CKtoB:=proc(a,b,c,d,e):[b, c, e, a, a, c, d]:end: #GBCZ3ck(a,b,c,d,e): A floating-point approximation to the constant #IntGBZ3(b, c, e, a, a, c, d) supposed to be good for the number of digits #It also returns [b, c, e, a, a, c, d] #Try: #GBCZ3ck(0,0,0,0,0); GBCZ3ck:=proc(a,b,c,d,e) local lu,K,RE: RE:=FindRelGZ3(b, c, e, a, a, c, d): if RE=FAIL then RETURN(FAIL): fi: lu:=GBCZ31ck(a,b,c,d,e,100): if lu=FAIL then RETURN(FAIL): fi: if lu[2]<10^(-3*Digits) then RETURN([lu[1], [b, c, e, a, a, c, d]]): fi: for K from 150 to 20000 by 50 do lu:=GBCZ31ck(a,b,c,d,e,K): if lu<>FAIL and lu[2]<10^(-3*Digits) then RETURN([lu[1], [b, c, e, a, a, c, d]]): fi: od: FAIL: end: #deltSeqZ3(a,b,c,d,e,,K): the sequence of empirial deltas for the sequence approximating #IntGBZ3ck(b, c, e, a, a, c, d,0); It also returns [b, c, e, a, a, c, d] #K must be at least 10. Try: #deltSeqZ3(0,0,0,0,0,100); deltSeqZ3:=proc(a,b,c,d,e,K) local c1,gu,ku,i,lu,RE,N,n,ope: if K<10 then print(K, `should have been at least 10`): RETURN(FAIL): fi: ope:=OPEZ3(a,b,c,d,e,n,N): lu:=ope[2]: ope:=ope[1]: RE:=FindRelGZ3(op(lu)): if RE=FAIL then RETURN(FAIL): fi: c1:=GBCZ3ck(a,b,c,d,e)[1]: if c1=FAIL then RETURN(FAIL): fi: gu:=AppxSeq1(ope,n,N,RE,K): ku:=[]: for i from 11 to nops(gu) while abs(gu[i]-c)>10^(-Digits) do ku:=[op(ku),delt(gu[i],c1)]: od: [ku,lu]: end: #IntGBZ3(a1,a2,b1,b2,c1,c2,d,n): #int(int(int(x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(-1)*(1-z)^(c2)/(1-z+x*y*z)^(d+1)*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n,z=0..1),x=0..1),y=0..1)): #Try: #IntGBZ3(0,0,0,0,0,0,0); IntGBZ3:=proc(a1,a2,b1,b2,c1,c2,d,n) local x,y,z,lu1,lu2: lu1:=int(int(int( x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(c1)*(1-z)^(c2)/(1-z+x*y*z)^(d+1)*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n,z=0..1),x=0..1),y=0..1): lu2:=int(int(int( x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^(c1)*(1-z)^(c2)/(1-z+x*y*z)^(d),z=0..1),x=0..1),y=0..1): lu1/lu2: end: IntGBZ3ck:=proc(a,b,c,d,e,n): IntGBZ3(b, c, e, a, a, c, d,n): end: #Khalek(Ope3,Ope2,n,N): Given a monic third-order operator Ope3 in n and the shift operator N #and a second-oder one, finds the operators Ope1a Ope1b such that #Ope3=Ope1a*Ope2+Ope1b. If Ope3 is a left multiple of Ope2, Ope1b should be zero. #Try #Khalek(N^3-1,N^2-1,n,N); Khalek:=proc(Ope3,Ope2,n,N) local gu,lu,Ope1a,Ope1b,i: gu:=expand(Ope3-subs(n=n+1,Ope2)*N): gu:=add(factor(coeff(gu,N,i))*N^i,i=0..2): lu:=coeff(gu,N,2): Ope1b:=gu-lu*Ope2: Ope1b:=add(factor(coeff(Ope1b,N,i))*N^i,i=0..degree(Ope1b,N)): Ope1a:=N+lu: [Ope1a,Ope1b]: end: #Kefel(ope1,ope2,n,N): ope1*ope2 Kefel:=proc(ope1,ope2,n,N) local gu,i: gu:=0: for i from 0 to degree(ope1,N) do gu:=gu+coeff(ope1,N,i)*N^i*subs(n=n+i,ope2): od: gu:=add(factor(coeff(gu,N,i))*N^i,i=0..degree(gu,N)): end: #NorOp(ope,n,N): normalizes the operator ope NorOp:=proc(ope,N) local i,ope1: ope1:=ope/coeff(ope,N,degree(ope,N)): add(factor(coeff(ope1,N,i))*N^i,i=0..degree(ope1,N)): end: ###Start Christoph's Koutschan operator ###End Christoph's Koutschan operator #OPEZ3(a,b,c,d,e,n,Sn): The 2nd-order linear recurrence operator in n and the shift operator Sn annnihilating the #double integral over [0,1]x[0,1] of the function # x^a1*(1-x)^a2*y^b1*(1-y)^b2*z^c1*(1-z)^c2*(x*(1-x)*y*(1-y)*z*(1-z))^n/(1-z+x*y*z)^(n+d+1) # parameters: [a1,a2,b1,b2,c1,c2,d]=[b, c, e, a, a, c, d] #It also returns [b, c, e, a, a, c, d], so the output is a pair [ope, ListOfLength7]. Try: #OPEZ3(0,0,0,0,0,n,Sn); OPEZ3:=proc(a,b,c,d,e,n,Sn) local ope: ope:=(-2+a-b-c-n)*(2+a+c-d+n)*(2+b+c-d+n)*(2+d+n)\ *(2+b+c-e+n)*(2+e+n)*(2+a-b+e+n)*(2+a-d+e+n)*(12+15*a+5*a^2+15*b+18*a*b+6*a^2*b+\ 5*b^2+6*a*b^2+2*a^2*b^2+30*c+32*a*c+9*a^2*c+28*b*c+27*a*b*c+7*a^2*b*c+6*b^2*c+5*\ a*b^2*c+a^2*b^2*c+28*c^2+24*a*c^2+5*a^2*c^2+18*b*c^2+13*a*b*c^2+2*a^2*b*c^2+2*b^\ 2*c^2+a*b^2*c^2+12*c^3+8*a*c^3+a^2*c^3+4*b*c^3+2*a*b*c^3+2*c^4+a*c^4-15*d-14*a*d\ -3*a^2*d-14*b*d-12*a*b*d-2*a^2*b*d-3*b^2*d-2*a*b^2*d-28*c*d-21*a*c*d-3*a^2*c*d-1\ 8*b*c*d-12*a*b*c*d-a^2*b*c*d-2*b^2*c*d-a*b^2*c*d-18*c^2*d-11*a*c^2*d-a^2*c^2*d-6\ *b*c^2*d-3*a*b*c^2*d-4*c^3*d-2*a*c^3*d+5*d^2+3*a*d^2+3*b*d^2+2*a*b*d^2+6*c*d^2+3\ *a*c*d^2+2*b*c*d^2+a*b*c*d^2+2*c^2*d^2+a*c^2*d^2+15*e+14*a*e+3*a^2*e+18*b*e+15*a\ *b*e+3*a^2*b*e+6*b^2*e+5*a*b^2*e+a^2*b^2*e+32*c*e+24*a*c*e+4*a^2*c*e+27*b*c*e+16\ *a*b*c*e+2*a^2*b*c*e+5*b^2*c*e+2*a*b^2*c*e+24*c^2*e+12*a*c^2*e+a^2*c^2*e+13*b*c^\ 2*e+4*a*b*c^2*e+b^2*c^2*e+8*c^3*e+2*a*c^3*e+2*b*c^3*e+c^4*e-14*d*e-9*a*d*e-a^2*d\ *e-12*b*d*e-8*a*b*d*e-a^2*b*d*e-2*b^2*d*e-a*b^2*d*e-21*c*d*e-8*a*c*d*e-12*b*c*d*\ e-4*a*b*c*d*e-b^2*c*d*e-11*c^2*d*e-2*a*c^2*d*e-3*b*c^2*d*e-2*c^3*d*e+3*d^2*e+a*d\ ^2*e+2*b*d^2*e+a*b*d^2*e+3*c*d^2*e+b*c*d^2*e+c^2*d^2*e+5*e^2+3*a*e^2+6*b*e^2+3*a\ *b*e^2+2*b^2*e^2+a*b^2*e^2+9*c*e^2+4*a*c*e^2+7*b*c*e^2+2*a*b*c*e^2+b^2*c*e^2+5*c\ ^2*e^2+a*c^2*e^2+2*b*c^2*e^2+c^3*e^2-3*d*e^2-a*d*e^2-2*b*d*e^2-a*b*d*e^2-3*c*d*e\ ^2-b*c*d*e^2-c^2*d*e^2+60*n+60*a*n+15*a^2*n+60*b*n+54*a*b*n+12*a^2*b*n+15*b^2*n+\ 12*a*b^2*n+2*a^2*b^2*n+120*c*n+96*a*c*n+18*a^2*c*n+84*b*c*n+54*a*b*c*n+7*a^2*b*c\ *n+12*b^2*c*n+5*a*b^2*c*n+84*c^2*n+48*a*c^2*n+5*a^2*c^2*n+36*b*c^2*n+13*a*b*c^2*\ n+2*b^2*c^2*n+24*c^3*n+8*a*c^3*n+4*b*c^3*n+2*c^4*n-60*d*n-42*a*d*n-6*a^2*d*n-42*\ b*d*n-24*a*b*d*n-2*a^2*b*d*n-6*b^2*d*n-2*a*b^2*d*n-84*c*d*n-42*a*c*d*n-3*a^2*c*d\ *n-36*b*c*d*n-12*a*b*c*d*n-2*b^2*c*d*n-36*c^2*d*n-11*a*c^2*d*n-6*b*c^2*d*n-4*c^3\ *d*n+15*d^2*n+6*a*d^2*n+6*b*d^2*n+2*a*b*d^2*n+12*c*d^2*n+3*a*c*d^2*n+2*b*c*d^2*n\ +2*c^2*d^2*n+60*e*n+42*a*e*n+6*a^2*e*n+54*b*e*n+30*a*b*e*n+3*a^2*b*e*n+12*b^2*e*\ n+5*a*b^2*e*n+96*c*e*n+48*a*c*e*n+4*a^2*c*e*n+54*b*c*e*n+16*a*b*c*e*n+5*b^2*c*e*\ n+48*c^2*e*n+12*a*c^2*e*n+13*b*c^2*e*n+8*c^3*e*n-42*d*e*n-18*a*d*e*n-a^2*d*e*n-2\ 4*b*d*e*n-8*a*b*d*e*n-2*b^2*d*e*n-42*c*d*e*n-8*a*c*d*e*n-12*b*c*d*e*n-11*c^2*d*e\ *n+6*d^2*e*n+a*d^2*e*n+2*b*d^2*e*n+3*c*d^2*e*n+15*e^2*n+6*a*e^2*n+12*b*e^2*n+3*a\ *b*e^2*n+2*b^2*e^2*n+18*c*e^2*n+4*a*c*e^2*n+7*b*c*e^2*n+5*c^2*e^2*n-6*d*e^2*n-a*\ d*e^2*n-2*b*d*e^2*n-3*c*d*e^2*n+120*n^2+90*a*n^2+15*a^2*n^2+90*b*n^2+54*a*b*n^2+\ 6*a^2*b*n^2+15*b^2*n^2+6*a*b^2*n^2+180*c*n^2+96*a*c*n^2+9*a^2*c*n^2+84*b*c*n^2+2\ 7*a*b*c*n^2+6*b^2*c*n^2+84*c^2*n^2+24*a*c^2*n^2+18*b*c^2*n^2+12*c^3*n^2-90*d*n^2\ -42*a*d*n^2-3*a^2*d*n^2-42*b*d*n^2-12*a*b*d*n^2-3*b^2*d*n^2-84*c*d*n^2-21*a*c*d*\ n^2-18*b*c*d*n^2-18*c^2*d*n^2+15*d^2*n^2+3*a*d^2*n^2+3*b*d^2*n^2+6*c*d^2*n^2+90*\ e*n^2+42*a*e*n^2+3*a^2*e*n^2+54*b*e*n^2+15*a*b*e*n^2+6*b^2*e*n^2+96*c*e*n^2+24*a\ *c*e*n^2+27*b*c*e*n^2+24*c^2*e*n^2-42*d*e*n^2-9*a*d*e*n^2-12*b*d*e*n^2-21*c*d*e*\ n^2+3*d^2*e*n^2+15*e^2*n^2+3*a*e^2*n^2+6*b*e^2*n^2+9*c*e^2*n^2-3*d*e^2*n^2+120*n\ ^3+60*a*n^3+5*a^2*n^3+60*b*n^3+18*a*b*n^3+5*b^2*n^3+120*c*n^3+32*a*c*n^3+28*b*c*\ n^3+28*c^2*n^3-60*d*n^3-14*a*d*n^3-14*b*d*n^3-28*c*d*n^3+5*d^2*n^3+60*e*n^3+14*a\ *e*n^3+18*b*e*n^3+32*c*e*n^3-14*d*e*n^3+5*e^2*n^3+60*n^4+15*a*n^4+15*b*n^4+30*c*\ n^4-15*d*n^4+15*e*n^4+12*n^5)*Sn^2+(44928+106416*a+100632*a^2+47144*a^3+10920*a^\ 4+1000*a^5+106416*b+248688*a*b+233648*a^2*b+109224*a^3*b+25328*a^4*b+2328*a^5*b+\ 100632*b^2+233648*a*b^2+219252*a^2*b^2+102692*a^3*b^2+23916*a^4*b^2+2212*a^5*b^2\ +47144*b^3+109224*a*b^3+102692*a^2*b^3+48312*a^3*b^3+11324*a^4*b^3+1056*a^5*b^3+\ 10920*b^4+25328*a*b^4+23916*a^2*b^4+11324*a^3*b^4+2676*a^4*b^4+252*a^5*b^4+1000*\ b^5+2328*a*b^5+2212*a^2*b^5+1056*a^3*b^5+252*a^4*b^5+24*a^5*b^5+212832*c+473664*\ a*c+421188*a^2*c+185676*a^3*c+40492*a^4*c+3492*a^5*c+449952*b*c+980700*a*b*c+859\ 542*a^2*b*c+374942*a^3*b*c+81138*a^4*b*c+6958*a^5*b*c+375080*b^2*c+804126*a*b^2*\ c+696505*a^2*b^2*c+300993*a^3*b^2*c+64631*a^4*b^2*c+5505*a^5*b^2*c+152904*b^3*c+\ 322906*a*b^3*c+276357*a^2*b^3*c+118146*a^3*b^3*c+25107*a^4*b^3*c+2116*a^5*b^3*c+\ 30328*b^4*c+63042*a*b^4*c+53175*a^2*b^4*c+22395*a^3*b^4*c+4681*a^4*b^4*c+387*a^5\ *b^4*c+2328*b^5*c+4746*a*b^5*c+3921*a^2*b^5*c+1612*a^3*b^5*c+327*a^4*b^5*c+26*a^\ 5*b^5*c+449952*c^2+934592*a*c^2+774426*a^2*c^2+317646*a^3*c^2+64326*a^4*c^2+5138\ *a^5*c^2+842376*b*c^2+1701618*a*b*c^2+1378298*a^2*b*c^2+554076*a^3*b*c^2+110118*\ a^4*b*c^2+8634*a^5*b*c^2+612444*b^2*c^2+1205179*a*b^2*c^2+953829*a^2*b^2*c^2+374\ 865*a^3*b^2*c^2+72771*a^4*b^2*c^2+5560*a^5*b^2*c^2+214004*b^3*c^2+409581*a*b^3*c\ ^2+315452*a^2*b^3*c^2+120405*a^3*b^3*c^2+22607*a^4*b^3*c^2+1659*a^5*b^3*c^2+3555\ 6*b^4*c^2+65845*a*b^4*c^2+48927*a^2*b^4*c^2+17905*a^3*b^4*c^2+3189*a^4*b^4*c^2+2\ 18*a^5*b^4*c^2+2212*b^5*c^2+3921*a*b^5*c^2+2764*a^2*b^5*c^2+945*a^3*b^5*c^2+153*\ a^4*b^5*c^2+9*a^5*b^5*c^2+561584*c^3+1083168*a*c^3+829881*a^2*c^3+313371*a^3*c^3\ +58111*a^4*c^3+4221*a^5*c^3+919080*b*c^3+1712510*a*b*c^3+1271277*a^2*b*c^3+46530\ 0*a^3*b*c^3+83499*a^4*b*c^3+5846*a^5*b*c^3+572736*b^2*c^3+1029828*a*b^2*c^3+7377\ 97*a^2*b^2*c^3+259869*a^3*b^2*c^3+44616*a^4*b^2*c^3+2958*a^5*b^2*c^3+167256*b^3*\ c^3+288770*a*b^3*c^3+197883*a^2*b^3*c^3+66156*a^3*b^3*c^3+10638*a^4*b^3*c^3+645*\ a^5*b^3*c^3+22384*b^4*c^3+36720*a*b^4*c^3+23665*a^2*b^4*c^3+7317*a^3*b^4*c^3+105\ 6*a^4*b^4*c^3+54*a^5*b^4*c^3+1056*b^5*c^3+1612*a*b^5*c^3+945*a^2*b^5*c^3+256*a^3\ *b^5*c^3+30*a^4*b^5*c^3+a^5*b^5*c^3+459540*c^4+820073*a*c^4+576993*a^2*c^4+19855\ 7*a^3*c^4+33231*a^4*c^4+2150*a^5*c^4+645100*b*c^4+1104747*a*b*c^4+745706*a^2*b*c\ ^4+245397*a^3*b*c^4+39007*a^4*b*c^4+2367*a^5*b*c^4+335628*b^2*c^4+549359*a*b^2*c\ ^4+352803*a^2*b^2*c^4+109519*a^3*b^2*c^4+16176*a^4*b^2*c^4+887*a^5*b^2*c^4+78740\ *b^3*c^4+122073*a*b^3*c^4+73423*a^2*b^3*c^4+20970*a^3*b^3*c^4+2760*a^4*b^3*c^4+1\ 26*a^5*b^3*c^4+7956*b^4*c^4+11481*a*b^4*c^4+6279*a^2*b^4*c^4+1571*a^3*b^4*c^4+16\ 8*a^4*b^4*c^4+5*a^5*b^4*c^4+252*b^5*c^4+327*a*b^5*c^4+153*a^2*b^5*c^4+30*a^3*b^5\ *c^4+2*a^4*b^5*c^4+258040*c^5+424950*a*c^5+272790*a^2*c^5+84630*a^3*c^5+12569*a^\ 4*c^5+705*a^5*c^5+302400*b*c^5+474592*a*b*c^5+288852*a^2*b*c^5+84196*a^3*b*c^5+1\ 1559*a^4*b*c^5+581*a^5*b*c^5+126248*b^2*c^5+187446*a*b^2*c^5+106643*a^2*b^2*c^5+\ 28521*a^3*b^2*c^5+3474*a^4*b^2*c^5+144*a^5*b^2*c^5+22320*b^3*c^5+30912*a*b^3*c^5\ +16017*a^2*b^3*c^5+3757*a^3*b^3*c^5+372*a^4*b^3*c^5+10*a^5*b^3*c^5+1512*b^4*c^5+\ 1902*a*b^4*c^5+853*a^2*b^4*c^5+159*a^3*b^4*c^5+10*a^4*b^4*c^5+24*b^5*c^5+26*a*b^\ 5*c^5+9*a^2*b^5*c^5+a^3*b^5*c^5+100800*c^6+152904*a*c^6+88926*a^2*c^6+24540*a^3*\ c^6+3159*a^4*c^6+147*a^5*c^6+94752*b*c^6+135948*a*b*c^6+73857*a^2*b*c^6+18675*a^\ 3*b*c^6+2127*a^4*b*c^6+81*a^5*b*c^6+29784*b^2*c^6+39970*a*b^2*c^6+19827*a^2*b^2*\ 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2*c^3*e^2+424460*a^2*b^2*c^3*e^2+89124*a^3*b^2*c^3*e^2+7878*a^4*b^2*c^3*e^2+198*\ a^5*b^2*c^3*e^2+197883*b^3*c^3*e^2+226116*a*b^3*c^3*e^2+96096*a^2*b^3*c^3*e^2+17\ 846*a^3*b^3*c^3*e^2+1302*a^4*b^3*c^3*e^2+22*a^5*b^3*c^3*e^2+23665*b^4*c^3*e^2+24\ 393*a*b^4*c^3*e^2+9040*a^2*b^4*c^3*e^2+1356*a^3*b^4*c^3*e^2+64*a^4*b^4*c^3*e^2+9\ 45*b^5*c^3*e^2+830*a*b^5*c^3*e^2+240*a^2*b^5*c^3*e^2+22*a^3*b^5*c^3*e^2+576993*c\ ^4*e^2+723133*a*c^4*e^2+337722*a^2*c^4*e^2+70881*a^3*c^4*e^2+6285*a^4*c^4*e^2+16\ 1*a^5*c^4*e^2+745706*b*c^4*e^2+860787*a*b*c^4*e^2+367670*a^2*b*c^4*e^2+69243*a^3\ *b*c^4*e^2+5291*a^4*b*c^4*e^2+105*a^5*b*c^4*e^2+352803*b^2*c^4*e^2+372138*a*b^2*\ c^4*e^2+142998*a^2*b^2*c^4*e^2+23384*a^3*b^2*c^4*e^2+1434*a^4*b^2*c^4*e^2+18*a^5\ *b^2*c^4*e^2+73423*b^3*c^4*e^2+69219*a*b^3*c^4*e^2+22942*a^2*b^3*c^4*e^2+2988*a^\ 3*b^3*c^4*e^2+116*a^4*b^3*c^4*e^2+6279*b^4*c^4*e^2+5036*a*b^4*c^4*e^2+1302*a^2*b\ ^4*c^4*e^2+104*a^3*b^4*c^4*e^2+153*b^5*c^4*e^2+93*a*b^5*c^4*e^2+14*a^2*b^5*c^4*e\ 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*a^3*b*c^7*e^2+2058*b^2*c^7*e^2+1074*a*b^2*c^7*e^2+128*a^2*b^2*c^7*e^2+90*b^3*c^\ 7*e^2+30*a*b^3*c^7*e^2+2850*c^8*e^2+1805*a*c^8*e^2+330*a^2*c^8*e^2+16*a^3*c^8*e^\ 2+1117*b*c^8*e^2+561*a*b*c^8*e^2+62*a^2*b*c^8*e^2+90*b^2*c^8*e^2+30*a*b^2*c^8*e^\ 2+241*c^9*e^2+117*a*c^9*e^2+12*a^2*c^9*e^2+45*b*c^9*e^2+15*a*b*c^9*e^2+9*c^10*e^\ 2+3*a*c^10*e^2-187540*d*e^2-306222*a*d*e^2-188580*a^2*d*e^2-53790*a^3*d*e^2-6896\ *a^4*d*e^2-300*a^5*d*e^2-383088*b*d*e^2-614689*a*b*d*e^2-374838*a^2*b*d*e^2-1069\ 23*a^3*b*d*e^2-13962*a^4*b*d*e^2-648*a^5*b*d*e^2-308076*b^2*d*e^2-487767*a*b^2*d\ *e^2-293992*a^2*b^2*d*e^2-83109*a^3*b^2*d*e^2-10827*a^4*b^2*d*e^2-513*a^5*b^2*d*\ e^2-120300*b^3*d*e^2-187917*a*b^3*d*e^2-111264*a^2*b^3*d*e^2-30758*a^3*b^3*d*e^2\ -3900*a^4*b^3*d*e^2-181*a^5*b^3*d*e^2-22516*b^4*d*e^2-34485*a*b^4*d*e^2-19726*a^\ 2*b^4*d*e^2-5157*a^3*b^4*d*e^2-596*a^4*b^4*d*e^2-24*a^5*b^4*d*e^2-1584*b^5*d*e^2\ -2336*a*b^5*d*e^2-1236*a^2*b^5*d*e^2-275*a^3*b^5*d*e^2-21*a^4*b^5*d*e^2-689310*c\ *d*e^2-1020042*a*c*d*e^2-563676*a^2*c*d*e^2-141600*a^3*c*d*e^2-15426*a^4*c*d*e^2\ -526*a^5*c*d*e^2-1230841*b*c*d*e^2-1769499*a*b*c*d*e^2-955130*a^2*b*c*d*e^2-2361\ 12*a^3*b*c*d*e^2-25765*a^4*b*c*d*e^2-939*a^5*b*c*d*e^2-848667*b^2*c*d*e^2-118526\ 3*a*b^2*c*d*e^2-619950*a^2*b^2*c*d*e^2-147907*a^3*b^2*c*d*e^2-15507*a^4*b^2*c*d*\ e^2-549*a^5*b^2*c*d*e^2-277981*b^3*c*d*e^2-375342*a*b^3*c*d*e^2-187628*a^2*b^3*c\ *d*e^2-42051*a^3*b^3*c*d*e^2-4011*a^4*b^3*c*d*e^2-123*a^5*b^3*c*d*e^2-42405*b^4*\ c*d*e^2-54680*a*b^4*c*d*e^2-25422*a^2*b^4*c*d*e^2-5067*a^3*b^4*c*d*e^2-387*a^4*b\ ^4*c*d*e^2-7*a^5*b^4*c*d*e^2-2336*b^5*c*d*e^2-2808*a*b^5*c*d*e^2-1154*a^2*b^5*c*\ d*e^2-183*a^3*b^5*c*d*e^2-8*a^4*b^5*c*d*e^2-1111345*c^2*d*e^2-1470570*a*c^2*d*e^\ 2-715486*a^2*c^2*d*e^2-153651*a^3*c^2*d*e^2-13426*a^4*c^2*d*e^2-300*a^5*c^2*d*e^\ 2-1711272*b*c^2*d*e^2-2173921*a*b*c^2*d*e^2-1017300*a^2*b*c^2*d*e^2-210774*a^3*b\ *c^2*d*e^2-18048*a^4*b*c^2*d*e^2-443*a^5*b*c^2*d*e^2-992839*b^2*c^2*d*e^2-120527\ 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5*b*c^6*n+104636*b^2*c^6*n+109950*a*b^2*c^6*n+40281*a^2*b^2*c^6*n+5907*a^3*b^2*c\ ^6*n+272*a^4*b^2*c^6*n+9996*b^3*c^6*n+8958*a*b^3*c^6*n+2538*a^2*b^3*c^6*n+224*a^\ 3*b^3*c^6*n+260*b^4*c^6*n+180*a*b^4*c^6*n+30*a^2*b^4*c^6*n+113856*c^7*n+130088*a\ *c^7*n+53688*a^2*c^7*n+9612*a^3*c^7*n+684*a^4*c^7*n+12*a^5*c^7*n+67272*b*c^7*n+6\ 8892*a*b*c^7*n+24340*a^2*b*c^7*n+3408*a^3*b*c^7*n+148*a^4*b*c^7*n+11424*b^2*c^7*\ n+10052*a*b^2*c^7*n+2772*a^2*b^2*c^7*n+236*a^3*b^2*c^7*n+520*b^3*c^7*n+360*a*b^3\ *c^7*n+60*a^2*b^3*c^7*n+16818*c^8*n+16866*a*c^8*n+5779*a^2*c^8*n+777*a^3*c^8*n+3\ 2*a^4*c^8*n+6426*b*c^8*n+5573*a*b*c^8*n+1503*a^2*b*c^8*n+124*a^3*b*c^8*n+520*b^2\ *c^8*n+360*a*b^2*c^8*n+60*a^2*b^2*c^8*n+1428*c^9*n+1224*a*c^9*n+324*a^2*c^9*n+26\ *a^3*c^9*n+260*b*c^9*n+180*a*b*c^9*n+30*a^2*b*c^9*n+52*c^10*n+36*a*c^10*n+6*a^2*\ c^10*n-923616*d*n-1776968*a*d*n-1339704*a^2*d*n-490564*a^3*d*n-86640*a^4*d*n-585\ 2*a^5*d*n-1776968*b*d*n-3327336*a*b*d*n-2439782*a^2*b*d*n-865710*a^3*b*d*n-14706\ 6*a^4*b*d*n-9426*a^5*b*d*n-1339704*b^2*d*n-2439782*a*b^2*d*n-1732050*a^2*b^2*d*n\ -590050*a^3*b^2*d*n-94818*a^4*b^2*d*n-5592*a^5*b^2*d*n-490564*b^3*d*n-865710*a*b\ ^3*d*n-590050*a^2*b^3*d*n-190008*a^3*b^3*d*n-28044*a^4*b^3*d*n-1428*a^5*b^3*d*n-\ 86640*b^4*d*n-147066*a*b^4*d*n-94818*a^2*b^4*d*n-28044*a^3*b^4*d*n-3570*a^4*b^4*\ d*n-130*a^5*b^4*d*n-5852*b^5*d*n-9426*a*b^5*d*n-5592*a^2*b^5*d*n-1428*a^3*b^5*d*\ n-130*a^4*b^5*d*n-3553936*c*d*n-6330708*a*c*d*n-4395519*a^2*c*d*n-1471845*a^3*c*\ d*n-235229*a^4*c*d*n-14139*a^5*c*d*n-6006744*b*c*d*n-10333908*a*b*c*d*n-6911610*\ a^2*b*c*d*n-2215656*a^3*b*c*d*n-335070*a^4*b*c*d*n-18644*a^5*b*c*d*n-3911474*b^2\ *c*d*n-6478125*a*b^2*c*d*n-4141289*a^2*b^2*c*d*n-1253532*a^3*b^2*c*d*n-175120*a^\ 4*b^2*c*d*n-8616*a^5*b^2*c*d*n-1212270*b^3*c*d*n-1919738*a*b^3*c*d*n-1158156*a^2\ *b^3*c*d*n-323820*a^3*b^3*c*d*n-40110*a^4*b^3*c*d*n-1588*a^5*b^3*c*d*n-176326*b^\ 4*c*d*n-263817*a*b^4*c*d*n-147034*a^2*b^4*c*d*n-36540*a^3*b^4*c*d*n-3697*a^4*b^4\ *c*d*n-90*a^5*b^4*c*d*n-9426*b^5*c*d*n-13052*a*b^5*c*d*n-6474*a^2*b^5*c*d*n-1328\ *a^3*b^5*c*d*n-90*a^4*b^5*c*d*n-6006744*c^2*d*n-9849863*a*c^2*d*n-6246126*a^2*c^\ 2*d*n-1891569*a^3*c^2*d*n-269526*a^4*c^2*d*n-14116*a^5*c^2*d*n-8791038*b*c^2*d*n\ -13806630*a*b*c^2*d*n-8339264*a^2*b*c^2*d*n-2381580*a^3*b*c^2*d*n-314314*a^4*b*c\ ^2*d*n-14730*a^5*b*c^2*d*n-4849020*b^2*c^2*d*n-7247957*a*b^2*c^2*d*n-4119192*a^2\ *b^2*c^2*d*n-1087004*a^3*b^2*c^2*d*n-128292*a^4*b^2*c^2*d*n-5019*a^5*b^2*c^2*d*n\ -1236834*b^3*c^2*d*n-1739892*a*b^3*c^2*d*n-912302*a^2*b^3*c^2*d*n-215196*a^3*b^3\ *c^2*d*n-21344*a^4*b^3*c^2*d*n-600*a^5*b^3*c^2*d*n-141948*b^4*c^2*d*n-184334*a*b\ ^4*c^2*d*n-86214*a^2*b^4*c^2*d*n-17107*a^3*b^4*c^2*d*n-1251*a^4*b^4*c^2*d*n-16*a\ ^5*b^4*c^2*d*n-5592*b^5*c^2*d*n-6474*a*b^5*c^2*d*n-2534*a^2*b^5*c^2*d*n-372*a^3*\ b^5*c^2*d*n-14*a^4*b^5*c^2*d*n-5860692*c^3*d*n-8800350*a*c^3*d*n-5053988*a^2*c^3\ *d*n-1367718*a^3*c^3*d*n-170839*a^4*c^3*d*n-7605*a^5*c^3*d*n-7273500*b*c^3*d*n-1\ 0360520*a*b*c^3*d*n-5587728*a^2*b*c^3*d*n-1397412*a^3*b*c^3*d*n-156780*a^4*b*c^3\ *d*n-5926*a^5*b*c^3*d*n-3301116*b^2*c^3*d*n-4414542*a*b^2*c^3*d*n-2194430*a^2*b^\ 2*c^3*d*n-491565*a^3*b^2*c^3*d*n-46854*a^4*b^2*c^3*d*n-1335*a^5*b^2*c^3*d*n-6635\ 40*b^3*c^3*d*n-818460*a*b^3*c^3*d*n-363432*a^2*b^3*c^3*d*n-69016*a^3*b^3*c^3*d*n\ -4998*a^4*b^3*c^3*d*n-78*a^5*b^3*c^3*d*n-56004*b^4*c^3*d*n-61770*a*b^4*c^3*d*n-2\ 3137*a^2*b^4*c^3*d*n-3327*a^3*b^4*c^3*d*n-136*a^4*b^4*c^3*d*n-1428*b^5*c^3*d*n-1\ 328*a*b^5*c^3*d*n-372*a^2*b^5*c^3*d*n-30*a^3*b^5*c^3*d*n-3636750*c^4*d*n-4973398\ *a*c^4*d*n-2561148*a^2*c^4*d*n-609785*a^3*c^4*d*n-65154*a^4*c^4*d*n-2365*a^5*c^4\ *d*n-3714840*b*c^4*d*n-4762980*a*b*c^4*d*n-2260336*a^2*b*c^4*d*n-482808*a^3*b*c^\ 4*d*n-44078*a^4*b*c^4*d*n-1230*a^5*b*c^4*d*n-1328010*b^2*c^4*d*n-1570884*a*b^2*c\ ^4*d*n-667134*a^2*b^2*c^4*d*n-121390*a^3*b^2*c^4*d*n-8538*a^4*b^2*c^4*d*n-138*a^\ 5*b^2*c^4*d*n-196140*b^3*c^4*d*n-208170*a*b^3*c^4*d*n-75064*a^2*b^3*c^4*d*n-1048\ 2*a^3*b^3*c^4*d*n-432*a^4*b^3*c^4*d*n-10710*b^4*c^4*d*n-9687*a*b^4*c^4*d*n-2661*\ a^2*b^4*c^4*d*n-216*a^3*b^4*c^4*d*n-130*b^5*c^4*d*n-90*a*b^5*c^4*d*n-14*a^2*b^5*\ c^4*d*n-1485936*c^5*d*n-1838652*a*c^5*d*n-837830*a^2*c^5*d*n-171390*a^3*c^5*d*n-\ 15001*a^4*c^5*d*n-405*a^5*c^5*d*n-1195488*b*c^5*d*n-1365924*a*b*c^5*d*n-558240*a\ ^2*b*c^5*d*n-97652*a^3*b*c^5*d*n-6636*a^4*b*c^5*d*n-106*a^5*b*c^5*d*n-313908*b^2\ *c^5*d*n-322962*a*b^2*c^5*d*n-112745*a^2*b^2*c^5*d*n-15279*a^3*b^2*c^5*d*n-620*a\ ^4*b^2*c^5*d*n-29988*b^3*c^5*d*n-26536*a*b^3*c^5*d*n-7164*a^2*b^3*c^5*d*n-578*a^\ 3*b^3*c^5*d*n-780*b^4*c^5*d*n-540*a*b^4*c^5*d*n-86*a^2*b^4*c^5*d*n-398496*c^6*d*\ n-442218*a*c^6*d*n-174798*a^2*c^6*d*n-29499*a^3*c^6*d*n-1935*a^4*c^6*d*n-30*a^5*\ c^6*d*n-235452*b*c^6*d*n-236124*a*b*c^6*d*n-80176*a^2*b*c^6*d*n-10560*a^3*b*c^6*\ d*n-418*a^4*b*c^6*d*n-39984*b^2*c^6*d*n-34766*a*b^2*c^6*d*n-9246*a^2*b^2*c^6*d*n\ -738*a^3*b^2*c^6*d*n-1820*b^3*c^6*d*n-1260*a*b^3*c^6*d*n-204*a^2*b^3*c^6*d*n-672\ 72*c^7*d*n-66036*a*c^7*d*n-21892*a^2*c^7*d*n-2808*a^3*c^7*d*n-108*a^4*c^7*d*n-25\ 704*b*c^7*d*n-22032*a*b*c^7*d*n-5784*a^2*b*c^7*d*n-456*a^3*b*c^7*d*n-2080*b^2*c^\ 7*d*n-1440*a*b^2*c^7*d*n-236*a^2*b^2*c^7*d*n-6426*c^8*d*n-5443*a*c^8*d*n-1413*a^\ 2*c^8*d*n-110*a^3*c^8*d*n-1170*b*c^8*d*n-810*a*b*c^8*d*n-134*a^2*b*c^8*d*n-260*c\ ^9*d*n-180*a*c^9*d*n-30*a^2*c^9*d*n+796268*d^2*n+1339704*a*d^2*n+859104*a^2*d^2*\ n+258396*a^3*d^2*n+35516*a^4*d^2*n+1692*a^5*d^2*n+1339704*b*d^2*n+2193266*a*b*d^\ 2*n+1357944*a^2*b*d^2*n+389862*a^3*b*d^2*n+50088*a^4*b*d^2*n+2128*a^5*b*d^2*n+85\ 9104*b^2*d^2*n+1357944*a*b^2*d^2*n+799802*a^2*b^2*d^2*n+213822*a^3*b^2*d^2*n+245\ 64*a^4*b^2*d^2*n+840*a^5*b^2*d^2*n+258396*b^3*d^2*n+389862*a*b^3*d^2*n+213822*a^\ 2*b^3*d^2*n+51252*a^3*b^3*d^2*n+4872*a^4*b^3*d^2*n+104*a^5*b^3*d^2*n+35516*b^4*d\ ^2*n+50088*a*b^4*d^2*n+24564*a^2*b^4*d^2*n+4872*a^3*b^4*d^2*n+312*a^4*b^4*d^2*n+\ 1692*b^5*d^2*n+2128*a*b^5*d^2*n+840*a^2*b^5*d^2*n+104*a^3*b^5*d^2*n+2679408*c*d^\ 2*n+4149003*a*c*d^2*n+2424510*a^2*c*d^2*n+655825*a^3*c*d^2*n+79362*a^4*c*d^2*n+3\ 192*a^5*c*d^2*n+3911474*b*c*d^2*n+5848671*a*b*c*d^2*n+3264233*a^2*b*c*d^2*n+8301\ 54*a^3*b*c*d^2*n+91785*a^4*b*c*d^2*n+3159*a^5*b*c*d^2*n+2133132*b^2*c*d^2*n+3049\ 224*a*b^2*c*d^2*n+1595646*a^2*b^2*c*d^2*n+369993*a^3*b^2*c*d^2*n+35358*a^4*b^2*c\ *d^2*n+913*a^5*b^2*c*d^2*n+531926*b^3*c*d^2*n+716508*a*b^3*c*d^2*n+342243*a^2*b^\ 3*c*d^2*n+68952*a^3*b^3*c*d^2*n+5153*a^4*b^3*c*d^2*n+72*a^5*b^3*c*d^2*n+58548*b^\ 4*c*d^2*n+72541*a*b^4*c*d^2*n+30150*a^2*b^4*c*d^2*n+4789*a^3*b^4*c*d^2*n+216*a^4\ *b^4*c*d^2*n+2128*b^5*c*d^2*n+2319*a*b^5*c*d^2*n+757*a^2*b^5*c*d^2*n+72*a^3*b^5*\ c*d^2*n+3911474*c^2*d^2*n+5557293*a*c^2*d^2*n+2944479*a^2*c^2*d^2*n+711840*a^3*c\ ^2*d^2*n+75397*a^4*c^2*d^2*n+2565*a^5*c^2*d^2*n+4849020*b*c^2*d^2*n+6593491*a*b*\ c^2*d^2*n+3290922*a^2*b*c^2*d^2*n+732835*a^3*b*c^2*d^2*n+68682*a^4*b*c^2*d^2*n+1\ 890*a^5*b*c^2*d^2*n+2182484*b^2*c^2*d^2*n+2803176*a*b^2*c^2*d^2*n+1286785*a^2*b^\ 2*c^2*d^2*n+253464*a^3*b^2*c^2*d^2*n+19498*a^4*b^2*c^2*d^2*n+363*a^5*b^2*c^2*d^2\ *n+431094*b^3*c^2*d^2*n+512871*a*b^3*c^2*d^2*n+208638*a^2*b^3*c^2*d^2*n+33918*a^\ 3*b^3*c^2*d^2*n+1839*a^4*b^3*c^2*d^2*n+14*a^5*b^3*c^2*d^2*n+35204*b^4*c^2*d^2*n+\ 37545*a*b^4*c^2*d^2*n+12652*a^2*b^4*c^2*d^2*n+1467*a^3*b^4*c^2*d^2*n+36*a^4*b^4*\ c^2*d^2*n+840*b^5*c^2*d^2*n+757*a*b^5*c^2*d^2*n+183*a^2*b^5*c^2*d^2*n+10*a^3*b^5\ *c^2*d^2*n+3232680*c^3*d^2*n+4200092*a*c^3*d^2*n+2004606*a^2*c^3*d^2*n+428771*a^\ 3*c^3*d^2*n+39186*a^4*c^3*d^2*n+1107*a^5*c^3*d^2*n+3301116*b*c^3*d^2*n+4059690*a\ *b*c^3*d^2*n+1791702*a^2*b*c^3*d^2*n+342603*a^3*b*c^3*d^2*n+26324*a^4*b*c^3*d^2*\ n+543*a^5*b*c^3*d^2*n+1172736*b^2*c^3*d^2*n+1340884*a*b^2*c^3*d^2*n+529254*a^2*b\ ^2*c^3*d^2*n+85212*a^3*b^2*c^3*d^2*n+4872*a^4*b^2*c^3*d^2*n+52*a^5*b^2*c^3*d^2*n\ +170788*b^3*c^3*d^2*n+176454*a*b^3*c^3*d^2*n+58840*a^2*b^3*c^3*d^2*n+7092*a^3*b^\ 3*c^3*d^2*n+220*a^4*b^3*c^3*d^2*n+9072*b^4*c^3*d^2*n+8054*a*b^4*c^3*d^2*n+2022*a\ ^2*b^4*c^3*d^2*n+134*a^3*b^4*c^3*d^2*n+104*b^5*c^3*d^2*n+72*a*b^5*c^3*d^2*n+10*a\ ^2*b^5*c^3*d^2*n+1650558*c^4*d^2*n+1952208*a*c^4*d^2*n+830772*a^2*c^4*d^2*n+1542\ 09*a^3*c^4*d^2*n+11719*a^4*c^4*d^2*n+252*a^5*c^4*d^2*n+1328010*b*c^4*d^2*n+14653\ 36*a*b*c^4*d^2*n+561576*a^2*b*c^4*d^2*n+88845*a^3*b*c^4*d^2*n+5157*a^4*b*c^4*d^2\ *n+62*a^5*b*c^4*d^2*n+346984*b^2*c^4*d^2*n+348024*a*b^2*c^4*d^2*n+114089*a^2*b^2\ *c^4*d^2*n+13845*a^3*b^2*c^4*d^2*n+464*a^4*b^2*c^4*d^2*n+32760*b^3*c^4*d^2*n+285\ 51*a*b^3*c^4*d^2*n+7233*a^2*b^3*c^4*d^2*n+512*a^3*b^3*c^4*d^2*n+832*b^4*c^4*d^2*\ n+576*a*b^4*c^4*d^2*n+86*a^2*b^4*c^4*d^2*n+531204*c^5*d^2*n+567974*a*c^5*d^2*n+2\ 11896*a^2*c^5*d^2*n+32892*a^3*c^5*d^2*n+1908*a^4*c^5*d^2*n+24*a^5*c^5*d^2*n+3139\ 08*b*c^5*d^2*n+306750*a*b*c^5*d^2*n+98857*a^2*b*c^5*d^2*n+11955*a^3*b*c^5*d^2*n+\ 412*a^4*b*c^5*d^2*n+53088*b^2*c^5*d^2*n+45500*a*b^2*c^5*d^2*n+11544*a^2*b^2*c^5*\ d^2*n+842*a^3*b^2*c^5*d^2*n+2392*b^3*c^5*d^2*n+1656*a*b^3*c^5*d^2*n+258*a^2*b^3*\ c^5*d^2*n+104636*c^6*d^2*n+99954*a*c^6*d^2*n+31713*a^2*c^6*d^2*n+3807*a^3*c^6*d^\ 2*n+132*a^4*c^6*d^2*n+39984*b*c^6*d^2*n+33778*a*b*c^6*d^2*n+8562*a^2*b*c^6*d^2*n\ +634*a^3*b*c^6*d^2*n+3224*b^2*c^6*d^2*n+2232*a*b^2*c^6*d^2*n+358*a^2*b^2*c^6*d^2\ *n+11424*c^7*d^2*n+9532*a*c^7*d^2*n+2412*a^2*c^7*d^2*n+180*a^3*c^7*d^2*n+2080*b*\ c^7*d^2*n+1440*a*b*c^7*d^2*n+236*a^2*b*c^7*d^2*n+520*c^8*d^2*n+360*a*c^8*d^2*n+6\ 0*a^2*c^8*d^2*n-338580*d^3*n-490564*a*d^3*n-258396*a^2*d^3*n-59824*a^3*d^3*n-561\ 6*a^4*d^3*n-132*a^5*d^3*n-490564*b*d^3*n-695478*a*b*d^3*n-352454*a^2*b*d^3*n-768\ 60*a^3*b*d^3*n-6542*a^4*b*d^3*n-126*a^5*b*d^3*n-258396*b^2*d^3*n-352454*a*b^2*d^\ 3*n-166818*a^2*b^2*d^3*n-32554*a^3*b^2*d^3*n-2268*a^4*b^2*d^3*n-26*a^5*b^2*d^3*n\ -59824*b^3*d^3*n-76860*a*b^3*d^3*n-32554*a^2*b^3*d^3*n-5208*a^3*b^3*d^3*n-234*a^\ 4*b^3*d^3*n-5616*b^4*d^3*n-6542*a*b^4*d^3*n-2268*a^2*b^4*d^3*n-234*a^3*b^4*d^3*n\ -132*b^5*d^3*n-126*a*b^5*d^3*n-26*a^2*b^5*d^3*n-981128*c*d^3*n-1301613*a*c*d^3*n\ 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2*d^3*n-18004*a^2*b^3*c^2*d^3*n-1716*a^3*b^3*c^2*d^3*n-30*a^4*b^3*c^2*d^3*n-2898\ *b^4*c^2*d^3*n-2511*a*b^4*c^2*d^3*n-549*a^2*b^4*c^2*d^3*n-24*a^3*b^4*c^2*d^3*n-2\ 6*b^5*c^2*d^3*n-18*a*b^5*c^2*d^3*n-2*a^2*b^5*c^2*d^3*n-824556*c^3*d^3*n-919290*a\ *c^3*d^3*n-355194*a^2*c^3*d^3*n-56799*a^3*c^3*d^3*n-3390*a^4*c^3*d^3*n-45*a^5*c^\ 3*d^3*n-663540*b*c^3*d^3*n-700900*a*b*c^3*d^3*n-245928*a^2*b*c^3*d^3*n-33464*a^3\ *b*c^3*d^3*n-1494*a^4*b*c^3*d^3*n-10*a^5*b*c^3*d^3*n-170788*b^2*c^3*d^3*n-166122\ *a*b^2*c^3*d^3*n-50012*a^2*b^2*c^3*d^3*n-5136*a^3*b^2*c^3*d^3*n-124*a^4*b^2*c^3*\ d^3*n-15540*b^3*c^3*d^3*n-13300*a*b^3*c^3*d^3*n-3084*a^2*b^3*c^3*d^3*n-176*a^3*b\ ^3*c^3*d^3*n-364*b^4*c^3*d^3*n-252*a*b^4*c^3*d^3*n-34*a^2*b^4*c^3*d^3*n-331770*c\ ^4*d^3*n-337774*a*c^4*d^3*n-116148*a^2*c^4*d^3*n-15867*a^3*c^4*d^3*n-747*a^4*c^4\ *d^3*n-6*a^5*c^4*d^3*n-196140*b*c^4*d^3*n-185490*a*b*c^4*d^3*n-55644*a^2*b*c^4*d\ ^3*n-5922*a^3*b*c^4*d^3*n-162*a^4*b*c^4*d^3*n-32760*b^2*c^4*d^3*n-27589*a*b^2*c^\ 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a^2*b*c^3*d*e^2*n-201228*a^3*b*c^3*d*e^2*n-8304*a^4*b*c^3*d*e^2*n-52*a^5*b*c^3*d\ *e^2*n-2194430*b^2*c^3*d*e^2*n-1803339*a*b^2*c^3*d*e^2*n-489160*a^2*b^2*c^3*d*e^\ 2*n-48216*a^3*b^2*c^3*d*e^2*n-1244*a^4*b^2*c^3*d*e^2*n-363432*b^3*c^3*d*e^2*n-25\ 6412*a*b^3*c^3*d*e^2*n-54192*a^2*b^3*c^3*d*e^2*n-3232*a^3*b^3*c^3*d*e^2*n-23137*\ b^4*c^3*d*e^2*n-12717*a*b^4*c^3*d*e^2*n-1600*a^2*b^4*c^3*d*e^2*n-372*b^5*c^3*d*e\ ^2*n-122*a*b^5*c^3*d*e^2*n-2561148*c^4*d*e^2*n-2160877*a*c^4*d*e^2*n-612456*a^2*\ c^4*d*e^2*n-65398*a^3*c^4*d*e^2*n-1995*a^4*c^4*d*e^2*n-2260336*b*c^4*d*e^2*n-170\ 0616*a*b*c^4*d*e^2*n-413278*a^2*b*c^4*d*e^2*n-35016*a^3*b*c^4*d*e^2*n-698*a^4*b*\ c^4*d*e^2*n-667134*b^2*c^4*d*e^2*n-428577*a*b^2*c^4*d*e^2*n-80940*a^2*b^2*c^4*d*\ e^2*n-4160*a^3*b^2*c^4*d*e^2*n-75064*b^3*c^4*d*e^2*n-37500*a*b^3*c^4*d*e^2*n-422\ 4*a^2*b^3*c^4*d*e^2*n-2661*b^4*c^4*d*e^2*n-798*a*b^4*c^4*d*e^2*n-14*b^5*c^4*d*e^\ 2*n-837830*c^5*d*e^2*n-581862*a*c^5*d*e^2*n-126938*a^2*c^5*d*e^2*n-9120*a^3*c^5*\ d*e^2*n-126*a^4*c^5*d*e^2*n-558240*b*c^5*d*e^2*n-329684*a*b*c^5*d*e^2*n-55848*a^\ 2*b*c^5*d*e^2*n-2452*a^3*b*c^5*d*e^2*n-112745*b^2*c^5*d*e^2*n-51753*a*b^2*c^5*d*\ e^2*n-5240*a^2*b^2*c^5*d*e^2*n-7164*b^3*c^5*d*e^2*n-1990*a*b^3*c^5*d*e^2*n-86*b^\ 4*c^5*d*e^2*n-174798*c^6*d*e^2*n-95605*a*c^6*d*e^2*n-14538*a^2*c^6*d*e^2*n-536*a\ ^3*c^6*d*e^2*n-80176*b*c^6*d*e^2*n-34104*a*b*c^6*d*e^2*n-3108*a^2*b*c^6*d*e^2*n-\ 9246*b^2*c^6*d*e^2*n-2402*a*b^2*c^6*d*e^2*n-204*b^3*c^6*d*e^2*n-21892*c^7*d*e^2*\ n-8688*a*c^7*d*e^2*n-712*a^2*c^7*d*e^2*n-5784*b*c^7*d*e^2*n-1416*a*b*c^7*d*e^2*n\ -236*b^2*c^7*d*e^2*n-1413*c^8*d*e^2*n-328*a*c^8*d*e^2*n-134*b*c^8*d*e^2*n-30*c^9\ *d*e^2*n+859104*d^2*e^2*n+1066566*a*d^2*e^2*n+480048*a^2*d^2*e^2*n+95508*a^3*d^2\ *e^2*n+8176*a^4*d^2*e^2*n+246*a^5*d^2*e^2*n+1357944*b*d^2*e^2*n+1664629*a*b*d^2*\ e^2*n+741096*a^2*b*d^2*e^2*n+146605*a^3*b*d^2*e^2*n+12582*a^4*b*d^2*e^2*n+376*a^\ 5*b*d^2*e^2*n+799802*b^2*d^2*e^2*n+954180*a*b^2*d^2*e^2*n+407440*a^2*b^2*d^2*e^2\ 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e^2*n+24629*a*b^4*c*d^2*e^2*n+5754*a^2*b^4*c*d^2*e^2*n+364*a^3*b^4*c*d^2*e^2*n+7\ 57*b^5*c*d^2*e^2*n+498*a*b^5*c*d^2*e^2*n+68*a^2*b^5*c*d^2*e^2*n+2944479*c^2*d^2*\ e^2*n+2703546*a*c^2*d^2*e^2*n+830872*a^2*c^2*d^2*e^2*n+95571*a^3*c^2*d^2*e^2*n+2\ 963*a^4*c^2*d^2*e^2*n-15*a^5*c^2*d^2*e^2*n+3290922*b*c^2*d^2*e^2*n+2888257*a*b*c\ ^2*d^2*e^2*n+841716*a^2*b*c^2*d^2*e^2*n+92314*a^3*b*c^2*d^2*e^2*n+3072*a^4*b*c^2\ *d^2*e^2*n+14*a^5*b*c^2*d^2*e^2*n+1286785*b^2*c^2*d^2*e^2*n+1046214*a*b^2*c^2*d^\ 2*e^2*n+269916*a^2*b^2*c^2*d^2*e^2*n+24240*a^3*b^2*c^2*d^2*e^2*n+542*a^4*b^2*c^2\ *d^2*e^2*n+208638*b^3*c^2*d^2*e^2*n+149918*a*b^3*c^2*d^2*e^2*n+30720*a^2*b^3*c^2\ *d^2*e^2*n+1684*a^3*b^3*c^2*d^2*e^2*n+12652*b^4*c^2*d^2*e^2*n+7329*a*b^4*c^2*d^2\ *e^2*n+908*a^2*b^4*c^2*d^2*e^2*n+183*b^5*c^2*d^2*e^2*n+68*a*b^5*c^2*d^2*e^2*n+20\ 04606*c^3*d^2*e^2*n+1546629*a*c^3*d^2*e^2*n+377184*a^2*c^3*d^2*e^2*n+30172*a^3*c\ ^3*d^2*e^2*n+324*a^4*c^3*d^2*e^2*n-8*a^5*c^3*d^2*e^2*n+1791702*b*c^3*d^2*e^2*n+1\ 285701*a*b*c^3*d^2*e^2*n+283956*a^2*b*c^3*d^2*e^2*n+20160*a^3*b*c^3*d^2*e^2*n+27\ 2*a^4*b*c^3*d^2*e^2*n+529254*b^2*c^3*d^2*e^2*n+336016*a*b^2*c^3*d^2*e^2*n+59712*\ a^2*b^2*c^3*d^2*e^2*n+2704*a^3*b^2*c^3*d^2*e^2*n+58840*b^3*c^3*d^2*e^2*n+30120*a\ *b^3*c^3*d^2*e^2*n+3280*a^2*b^3*c^3*d^2*e^2*n+2022*b^4*c^3*d^2*e^2*n+654*a*b^4*c\ ^3*d^2*e^2*n+10*b^5*c^3*d^2*e^2*n+830772*c^4*d^2*e^2*n+526299*a*c^4*d^2*e^2*n+97\ 346*a^2*c^4*d^2*e^2*n+4878*a^3*c^4*d^2*e^2*n-4*a^4*c^4*d^2*e^2*n+561576*b*c^4*d^\ 2*e^2*n+316328*a*b*c^4*d^2*e^2*n+48210*a^2*b*c^4*d^2*e^2*n+1700*a^3*b*c^4*d^2*e^\ 2*n+114089*b^2*c^4*d^2*e^2*n+52017*a*b^2*c^4*d^2*e^2*n+4936*a^2*b^2*c^4*d^2*e^2*\ n+7233*b^3*c^4*d^2*e^2*n+2090*a*b^3*c^4*d^2*e^2*n+86*b^4*c^4*d^2*e^2*n+211896*c^\ 5*d^2*e^2*n+106112*a*c^5*d^2*e^2*n+13596*a^2*c^5*d^2*e^2*n+328*a^3*c^5*d^2*e^2*n\ +98857*b*c^5*d^2*e^2*n+40389*a*b*c^5*d^2*e^2*n+3304*a^2*b*c^5*d^2*e^2*n+11544*b^\ 2*c^5*d^2*e^2*n+3022*a*b^2*c^5*d^2*e^2*n+258*b^3*c^5*d^2*e^2*n+31713*c^6*d^2*e^2\ 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2*a*c^3*d^4*e^2*n+384*a^2*c^3*d^4*e^2*n-24*a^3*c^3*d^4*e^2*n+13721*b*c^3*d^4*e^2\ *n+4929*a*b*c^3*d^4*e^2*n+248*a^2*b*c^3*d^4*e^2*n+1626*b^2*c^3*d^4*e^2*n+442*a*b\ ^2*c^3*d^4*e^2*n+34*b^3*c^3*d^4*e^2*n+6688*c^4*d^4*e^2*n+1794*a*c^4*d^4*e^2*n+28\ *a^2*c^4*d^4*e^2*n+1959*b*c^4*d^4*e^2*n+432*a*b*c^4*d^4*e^2*n+86*b^2*c^4*d^4*e^2\ *n+792*c^5*d^4*e^2*n+142*a*c^5*d^4*e^2*n+86*b*c^5*d^4*e^2*n+30*c^6*d^4*e^2*n-169\ 2*d^5*e^2*n-1064*a*d^5*e^2*n-246*a^2*d^5*e^2*n-26*a^3*d^5*e^2*n-1732*b*d^5*e^2*n\ -1290*a*b*d^5*e^2*n-298*a^2*b*d^5*e^2*n-24*a^3*b*d^5*e^2*n-462*b^2*d^5*e^2*n-361\ *a*b^2*d^5*e^2*n-75*a^2*b^2*d^5*e^2*n-4*a^3*b^2*d^5*e^2*n-26*b^3*d^5*e^2*n-18*a*\ b^3*d^5*e^2*n-2*a^2*b^3*d^5*e^2*n-2796*c*d^5*e^2*n-999*a*c*d^5*e^2*n-52*a^2*c*d^\ 5*e^2*n-3*a^3*c*d^5*e^2*n-2214*b*c*d^5*e^2*n-1066*a*b*c*d^5*e^2*n-102*a^2*b*c*d^\ 5*e^2*n-2*a^3*b*c*d^5*e^2*n-439*b^2*c*d^5*e^2*n-228*a*b^2*c*d^5*e^2*n-20*a^2*b^2\ *c*d^5*e^2*n-18*b^3*c*d^5*e^2*n-8*a*b^3*c*d^5*e^2*n-1998*c^2*d^5*e^2*n-459*a*c^2\ 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4*b^4*c^4*n^2+32438*a*b^4*c^4*n^2+8622*a^2*b^4*c^4*n^2+704*a^3*b^4*c^4*n^2+742*b\ ^5*c^4*n^2+459*a*b^5*c^4*n^2+69*a^2*b^5*c^4*n^2+3558040*c^5*n^2+4297536*a*c^5*n^\ 2+1935272*a^2*c^5*n^2+394920*a^3*c^5*n^2+34880*a^4*c^5*n^2+972*a^5*c^5*n^2+31006\ 80*b*c^5*n^2+3386548*a*b*c^5*n^2+1345878*a^2*b*c^5*n^2+231770*a^3*b*c^5*n^2+1574\ 1*a^4*b*c^5*n^2+261*a^5*b*c^5*n^2+921152*b^2*c^5*n^2+884106*a*b^2*c^5*n^2+294708\ *a^2*b^2*c^5*n^2+38736*a^3*b^2*c^5*n^2+1554*a^4*b^2*c^5*n^2+108720*b^3*c^5*n^2+8\ 6994*a*b^3*c^5*n^2+21915*a^2*b^3*c^5*n^2+1681*a^3*b^3*c^5*n^2+4452*b^4*c^5*n^2+2\ 664*a*b^4*c^5*n^2+384*a^2*b^4*c^5*n^2+36*b^5*c^5*n^2+12*a*b^5*c^5*n^2+1033560*c^\ 6*n^2+1090520*a*c^6*n^2+414252*a^2*c^6*n^2+67560*a^3*c^6*n^2+4302*a^4*c^6*n^2+66\ *a^5*c^6*n^2+690760*b*c^6*n^2+640044*a*b*c^6*n^2+203752*a^2*b*c^6*n^2+25335*a^3*\ b*c^6*n^2+951*a^4*b*c^6*n^2+144936*b^2*c^6*n^2+112214*a*b^2*c^6*n^2+27063*a^2*b^\ 2*c^6*n^2+1969*a^3*b^2*c^6*n^2+10388*b^3*c^6*n^2+6066*a*b^3*c^6*n^2+846*a^2*b^3*\ 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9124946*c^2*d*n^2-27644904*a*c^2*d*n^2-15239124*a^2*c^2*d*n^2-3938634*a^3*c^2*d*\ n^2-466654*a^4*c^2*d*n^2-19530*a^5*c^2*d*n^2-24716880*b*c^2*d*n^2-33643248*a*b*c\ ^2*d*n^2-17286480*a^2*b*c^2*d*n^2-4092639*a^3*b*c^2*d*n^2-430785*a^4*b*c^2*d*n^2\ -15098*a^5*b*c^2*d*n^2-11872398*b^2*c^2*d*n^2-15047838*a*b^2*c^2*d*n^2-7067121*a\ ^2*b^2*c^2*d*n^2-1483254*a^3*b^2*c^2*d*n^2-130793*a^4*b^2*c^2*d*n^2-3393*a^5*b^2\ *c^2*d*n^2-2588004*b^3*c^2*d*n^2-2997337*a*b^3*c^2*d*n^2-1245717*a^2*b^3*c^2*d*n\ ^2-218936*a^3*b^3*c^2*d*n^2-14400*a^4*b^3*c^2*d*n^2-200*a^5*b^3*c^2*d*n^2-247116\ *b^4*c^2*d*n^2-253188*a*b^4*c^2*d*n^2-87877*a^2*b^4*c^2*d*n^2-11529*a^3*b^4*c^2*\ d*n^2-417*a^4*b^4*c^2*d*n^2-7776*b^5*c^2*d*n^2-6640*a*b^5*c^2*d*n^2-1710*a^2*b^5\ *c^2*d*n^2-124*a^3*b^5*c^2*d*n^2-16477920*c^3*d*n^2-21448664*a*c^3*d*n^2-1048014\ 0*a^2*c^3*d*n^2-2350703*a^3*c^3*d*n^2-234009*a^4*c^3*d*n^2-7782*a^5*c^3*d*n^2-17\ 790200*b*c^3*d*n^2-21487680*a*b*c^3*d*n^2-9576224*a^2*b*c^3*d*n^2-1904058*a^3*b*\ 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c^8*d*n^2-471*a^2*c^8*d*n^2-810*b*c^8*d*n^2-270*a*b*c^8*d*n^2-180*c^9*d*n^2-60*a\ *c^9*d*n^2+2841390*d^2*n^2+4272261*a*d^2*n^2+2424510*a^2*d^2*n^2+637121*a^3*d^2*\ n^2+75168*a^4*d^2*n^2+2994*a^5*d^2*n^2+4272261*b*d^2*n^2+6163398*a*b*d^2*n^2+332\ 0345*a^2*b*d^2*n^2+815040*a^3*b*d^2*n^2+87292*a^4*b*d^2*n^2+2970*a^5*b*d^2*n^2+2\ 424510*b^2*d^2*n^2+3320345*a*b^2*d^2*n^2+1666152*a^2*b^2*d^2*n^2+370290*a^3*b^2*\ d^2*n^2+34056*a^4*b^2*d^2*n^2+874*a^5*b^2*d^2*n^2+637121*b^3*d^2*n^2+815040*a*b^\ 3*d^2*n^2+370290*a^2*b^3*d^2*n^2+70884*a^3*b^3*d^2*n^2+5062*a^4*b^3*d^2*n^2+72*a\ ^5*b^3*d^2*n^2+75168*b^4*d^2*n^2+87292*a*b^4*d^2*n^2+34056*a^2*b^4*d^2*n^2+5062*\ a^3*b^4*d^2*n^2+216*a^4*b^4*d^2*n^2+2994*b^5*d^2*n^2+2970*a*b^5*d^2*n^2+874*a^2*\ b^5*d^2*n^2+72*a^3*b^5*d^2*n^2+8544522*c*d^2*n^2+11669607*a*c*d^2*n^2+5936199*a^\ 2*c*d^2*n^2+1372896*a^3*c*d^2*n^2+138423*a^4*c*d^2*n^2+4455*a^5*c*d^2*n^2+110124\ 18*b*c*d^2*n^2+14271637*a*b*c*d^2*n^2+6783966*a^2*b*c*d^2*n^2+1432642*a^3*b*c*d^\ 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n^2-1023*a^4*d^3*e^2*n^2-18*a^5*d^3*e^2*n^2-736146*b*d^3*e^2*n^2-636459*a*b*d^3*\ e^2*n^2-183951*a^2*b*d^3*e^2*n^2-21106*a^3*b*d^3*e^2*n^2-882*a^4*b*d^3*e^2*n^2-8\ *a^5*b*d^3*e^2*n^2-288814*b^2*d^3*e^2*n^2-235251*a*b^2*d^3*e^2*n^2-60666*a^2*b^2\ *d^3*e^2*n^2-5616*a^3*b^2*d^3*e^2*n^2-143*a^4*b^2*d^3*e^2*n^2-45054*b^3*d^3*e^2*\ n^2-32538*a*b^3*d^3*e^2*n^2-6588*a^2*b^3*d^3*e^2*n^2-356*a^3*b^3*d^3*e^2*n^2-235\ 8*b^4*d^3*e^2*n^2-1341*a*b^4*d^3*e^2*n^2-153*a^2*b^4*d^3*e^2*n^2-18*b^5*d^3*e^2*\ n^2-6*a*b^5*d^3*e^2*n^2-1294002*c*d^3*e^2*n^2-906599*a*c*d^3*e^2*n^2-196182*a^2*\ c*d^3*e^2*n^2-14180*a^3*c*d^3*e^2*n^2-261*a^4*c*d^3*e^2*n^2-a^5*c*d^3*e^2*n^2-12\ 14087*b*c*d^3*e^2*n^2-822420*a*b*c*d^3*e^2*n^2-168572*a^2*b*c*d^3*e^2*n^2-11286*\ a^3*b*c*d^3*e^2*n^2-176*a^4*b*c*d^3*e^2*n^2-370413*b^2*c*d^3*e^2*n^2-226270*a*b^\ 2*c*d^3*e^2*n^2-37872*a^2*b^2*c*d^3*e^2*n^2-1620*a^3*b^2*c*d^3*e^2*n^2-41970*b^3\ *c*d^3*e^2*n^2-20772*a*b^3*c*d^3*e^2*n^2-2112*a^2*b^3*c*d^3*e^2*n^2-1431*b^4*c*d\ ^3*e^2*n^2-444*a*b^4*c*d^3*e^2*n^2-6*b^5*c*d^3*e^2*n^2-1091308*c^2*d^3*e^2*n^2-5\ 99400*a*c^2*d^3*e^2*n^2-89240*a^2*c^2*d^3*e^2*n^2-2970*a^3*c^2*d^3*e^2*n^2+16*a^\ 4*c^2*d^3*e^2*n^2-789615*b*c^2*d^3*e^2*n^2-402166*a*b*c^2*d^3*e^2*n^2-52884*a^2*\ b*c^2*d^3*e^2*n^2-1512*a^3*b*c^2*d^3*e^2*n^2-172428*b^2*c^2*d^3*e^2*n^2-72378*a*\ b^2*c^2*d^3*e^2*n^2-6064*a^2*b^2*c^2*d^3*e^2*n^2-12132*b^3*c^2*d^3*e^2*n^2-3216*\ a*b^3*c^2*d^3*e^2*n^2-183*b^4*c^2*d^3*e^2*n^2-484569*c^3*d^3*e^2*n^2-201508*a*c^\ 3*d^3*e^2*n^2-18900*a^2*c^3*d^3*e^2*n^2-168*a^3*c^3*d^3*e^2*n^2-249484*b*c^3*d^3\ *e^2*n^2-87858*a*b*c^3*d^3*e^2*n^2-5744*a^2*b*c^3*d^3*e^2*n^2-33606*b^2*c^3*d^3*\ e^2*n^2-7596*a*b^2*c^3*d^3*e^2*n^2-1028*b^3*c^3*d^3*e^2*n^2-117691*c^4*d^3*e^2*n\ ^2-34290*a*c^4*d^3*e^2*n^2-1590*a^2*c^4*d^3*e^2*n^2-37332*b*c^4*d^3*e^2*n^2-7180\ *a*b*c^4*d^3*e^2*n^2-2189*b^2*c^4*d^3*e^2*n^2-14463*c^5*d^3*e^2*n^2-2355*a*c^5*d\ ^3*e^2*n^2-2004*b*c^5*d^3*e^2*n^2-666*c^6*d^3*e^2*n^2+75168*d^4*e^2*n^2+51131*a*\ d^4*e^2*n^2+11322*a^2*d^4*e^2*n^2+1023*a^3*d^4*e^2*n^2+36*a^4*d^4*e^2*n^2+72427*\ b*d^4*e^2*n^2+50760*a*b*d^4*e^2*n^2+10949*a^2*b*d^4*e^2*n^2+828*a^3*b*d^4*e^2*n^\ 2+16*a^4*b*d^4*e^2*n^2+20988*b^2*d^4*e^2*n^2+13852*a*b^2*d^4*e^2*n^2+2511*a^2*b^\ 2*d^4*e^2*n^2+119*a^3*b^2*d^4*e^2*n^2+1962*b^3*d^4*e^2*n^2+1071*a*b^3*d^4*e^2*n^\ 2+117*a^2*b^3*d^4*e^2*n^2+36*b^4*d^4*e^2*n^2+12*a*b^4*d^4*e^2*n^2+123558*c*d^4*e\ ^2*n^2+60075*a*c*d^4*e^2*n^2+7386*a^2*c*d^4*e^2*n^2+207*a^3*c*d^4*e^2*n^2+2*a^4*\ c*d^4*e^2*n^2+92736*b*c*d^4*e^2*n^2+46094*a*b*c*d^4*e^2*n^2+5670*a^2*b*c*d^4*e^2\ *n^2+149*a^3*b*c*d^4*e^2*n^2+19738*b^2*c*d^4*e^2*n^2+8649*a*b^2*c*d^4*e^2*n^2+73\ 4*a^2*b^2*c*d^4*e^2*n^2+1215*b^3*c*d^4*e^2*n^2+354*a*b^3*c*d^4*e^2*n^2+12*b^4*c*\ d^4*e^2*n^2+83070*c^2*d^4*e^2*n^2+28679*a*c^2*d^4*e^2*n^2+1656*a^2*c^2*d^4*e^2*n\ ^2-19*a^3*c^2*d^4*e^2*n^2+44539*b*c^2*d^4*e^2*n^2+14733*a*b*c^2*d^4*e^2*n^2+794*\ a^2*b*c^2*d^4*e^2*n^2+5940*b^2*c^2*d^4*e^2*n^2+1373*a*b^2*c^2*d^4*e^2*n^2+165*b^\ 3*c^2*d^4*e^2*n^2+27786*c^3*d^4*e^2*n^2+6534*a*c^3*d^4*e^2*n^2+128*a^2*c^3*d^4*e\ ^2*n^2+9207*b*c^3*d^4*e^2*n^2+1643*a*b*c^3*d^4*e^2*n^2+542*b^2*c^3*d^4*e^2*n^2+4\ 482*c^4*d^4*e^2*n^2+598*a*c^4*d^4*e^2*n^2+653*b*c^4*d^4*e^2*n^2+264*c^5*d^4*e^2*\ n^2-2994*d^5*e^2*n^2-1485*a*d^5*e^2*n^2-253*a^2*d^5*e^2*n^2-18*a^3*d^5*e^2*n^2-2\ 403*b*d^5*e^2*n^2-1312*a*b*d^5*e^2*n^2-198*a^2*b*d^5*e^2*n^2-8*a^3*b*d^5*e^2*n^2\ -478*b^2*d^5*e^2*n^2-243*a*b^2*d^5*e^2*n^2-25*a^2*b^2*d^5*e^2*n^2-18*b^3*d^5*e^2\ *n^2-6*a*b^3*d^5*e^2*n^2-3888*c*d^5*e^2*n^2-1044*a*c*d^5*e^2*n^2-36*a^2*c*d^5*e^\ 2*n^2-a^3*c*d^5*e^2*n^2-2268*b*c*d^5*e^2*n^2-720*a*b*c*d^5*e^2*n^2-34*a^2*b*c*d^\ 5*e^2*n^2-297*b^2*c*d^5*e^2*n^2-76*a*b^2*c*d^5*e^2*n^2-6*b^3*c*d^5*e^2*n^2-2043*\ c^2*d^5*e^2*n^2-315*a*c^2*d^5*e^2*n^2+8*a^2*c^2*d^5*e^2*n^2-738*b*c^2*d^5*e^2*n^\ 2-118*a*b*c^2*d^5*e^2*n^2-43*b^2*c^2*d^5*e^2*n^2-477*c^3*d^5*e^2*n^2-41*a*c^3*d^\ 5*e^2*n^2-76*b*c^3*d^5*e^2*n^2-39*c^4*d^5*e^2*n^2+1084770*e^3*n^2+1386729*a*e^3*\ n^2+637121*a^2*e^3*n^2+126522*a^3*e^3*n^2+9945*a^4*e^3*n^2+189*a^5*e^3*n^2+19652\ 58*b*e^3*n^2+2334333*a*b*e^3*n^2+993330*a^2*b*e^3*n^2+180567*a^3*b*e^3*n^2+12636\ *a^4*b*e^3*n^2+198*a^5*b*e^3*n^2+1408854*b^2*e^3*n^2+1572408*a*b^2*e^3*n^2+62701\ 6*a^2*b^2*e^3*n^2+105525*a^3*b^2*e^3*n^2+6663*a^4*b^2*e^3*n^2+90*a^5*b^2*e^3*n^2\ +491148*b^3*e^3*n^2+515190*a*b^3*e^3*n^2+191169*a^2*b^3*e^3*n^2+29136*a^3*b^3*e^\ 3*n^2+1566*a^4*b^3*e^3*n^2+15*a^5*b^3*e^3*n^2+82164*b^4*e^3*n^2+80217*a*b^4*e^3*\ n^2+26977*a^2*b^4*e^3*n^2+3483*a^3*b^4*e^3*n^2+129*a^4*b^4*e^3*n^2+5184*b^5*e^3*\ n^2+4586*a*b^5*e^3*n^2+1305*a^2*b^5*e^3*n^2+115*a^3*b^5*e^3*n^2+3351987*c*e^3*n^\ 2+3822982*a*c*e^3*n^2+1551186*a^2*c*e^3*n^2+267276*a^3*c*e^3*n^2+17667*a^4*c*e^3\ *n^2+264*a^5*c*e^3*n^2+5152041*b*c*e^3*n^2+5347836*a*b*c*e^3*n^2+1956107*a^2*b*c\ *e^3*n^2+297000*a^3*b*c*e^3*n^2+16458*a^4*b*c*e^3*n^2+180*a^5*b*c*e^3*n^2+304585\ 2*b^2*c*e^3*n^2+2881078*a*b^2*c*e^3*n^2+947457*a^2*b^2*c*e^3*n^2+125132*a^3*b^2*\ c*e^3*n^2+5598*a^4*b^2*c*e^3*n^2+40*a^5*b^2*c*e^3*n^2+843846*b^3*c*e^3*n^2+71667\ 0*a*b^3*c*e^3*n^2+205140*a^2*b^3*c*e^3*n^2+21852*a^3*b^3*c*e^3*n^2+631*a^4*b^3*c\ *e^3*n^2+106137*b^4*c*e^3*n^2+77890*a*b^4*c*e^3*n^2+17829*a^2*b^4*c*e^3*n^2+1204\ *a^3*b^4*c*e^3*n^2+4586*b^5*c*e^3*n^2+2646*a*b^5*c*e^3*n^2+373*a^2*b^5*c*e^3*n^2\ +4380308*c^2*e^3*n^2+4333284*a*c^2*e^3*n^2+1495126*a^2*c^2*e^3*n^2+211788*a^3*c^\ 2*e^3*n^2+10818*a^4*c^2*e^3*n^2+108*a^5*c^2*e^3*n^2+5611590*b*c^2*e^3*n^2+491866\ 1*a*b*c^2*e^3*n^2+1472301*a^2*b*c^2*e^3*n^2+172944*a^3*b*c^2*e^3*n^2+6615*a^4*b*\ c^2*e^3*n^2+36*a^5*b*c^2*e^3*n^2+2665570*b^2*c^2*e^3*n^2+2042658*a*b^2*c^2*e^3*n\ ^2+515784*a^2*b^2*c^2*e^3*n^2+46980*a^3*b^2*c^2*e^3*n^2+1095*a^4*b^2*c^2*e^3*n^2\ +563877*b^3*c^2*e^3*n^2+362932*a*b^3*c^2*e^3*n^2+70902*a^2*b^3*c^2*e^3*n^2+3920*\ a^3*b^3*c^2*e^3*n^2+49907*b^4*c^2*e^3*n^2+24534*a*b^4*c^2*e^3*n^2+2863*a^2*b^4*c\ ^2*e^3*n^2+1305*b^5*c^2*e^3*n^2+373*a*b^5*c^2*e^3*n^2+3183408*c^3*e^3*n^2+263948\ 2*a*c^3*e^3*n^2+736326*a^2*c^3*e^3*n^2+79060*a^3*c^3*e^3*n^2+2682*a^4*c^3*e^3*n^\ 2+12*a^5*c^3*e^3*n^2+3308825*b*c^3*e^3*n^2+2342124*a*b*c^3*e^3*n^2+534144*a^2*b*\ c^3*e^3*n^2+42552*a^3*b*c^3*e^3*n^2+816*a^4*b*c^3*e^3*n^2+1212174*b^2*c^3*e^3*n^\ 2+705040*a*b^2*c^3*e^3*n^2+121104*a^2*b^2*c^3*e^3*n^2+5648*a^3*b^2*c^3*e^3*n^2+1\ 82904*b^3*c^3*e^3*n^2+79596*a*b^3*c^3*e^3*n^2+7976*a^2*b^3*c^3*e^3*n^2+10008*b^4\ *c^3*e^3*n^2+2494*a*b^4*c^3*e^3*n^2+115*b^5*c^3*e^3*n^2+1414882*c^4*e^3*n^2+9409\ 50*a*c^4*e^3*n^2+197546*a^2*c^4*e^3*n^2+14040*a^3*c^4*e^3*n^2+225*a^4*c^4*e^3*n^\ 2+1143981*b*c^4*e^3*n^2+613046*a*b*c^4*e^3*n^2+94410*a^2*b*c^4*e^3*n^2+3780*a^3*\ b*c^4*e^3*n^2+301793*b^2*c^4*e^3*n^2+118944*a*b^2*c^4*e^3*n^2+10456*a^2*b^2*c^4*\ e^3*n^2+28548*b^3*c^4*e^3*n^2+6361*a*b^3*c^4*e^3*n^2+704*b^4*c^4*e^3*n^2+394920*\ c^5*e^3*n^2+197700*a*c^5*e^3*n^2+27702*a^2*c^5*e^3*n^2+964*a^3*c^5*e^3*n^2+23177\ 0*b*c^5*e^3*n^2+84006*a*b*c^5*e^3*n^2+6572*a^2*b*c^5*e^3*n^2+38736*b^2*c^5*e^3*n\ ^2+7844*a*b^2*c^5*e^3*n^2+1681*b^3*c^5*e^3*n^2+67560*c^6*e^3*n^2+22770*a*c^6*e^3\ *n^2+1602*a^2*c^6*e^3*n^2+25335*b*c^6*e^3*n^2+4716*a*b*c^6*e^3*n^2+1969*b^2*c^6*\ e^3*n^2+6444*c^7*e^3*n^2+1112*a*c^7*e^3*n^2+1136*b*c^7*e^3*n^2+259*c^8*e^3*n^2-1\ 386729*d*e^3*n^2-1488649*a*d*e^3*n^2-557856*a^2*d*e^3*n^2-86709*a^3*d*e^3*n^2-50\ 31*a^4*d*e^3*n^2-66*a^5*d*e^3*n^2-2119926*b*d*e^3*n^2-2151432*a*b*d*e^3*n^2-7640\ 05*a^2*b*d*e^3*n^2-113094*a^3*b*d*e^3*n^2-6384*a^4*b*d*e^3*n^2-90*a^5*b*d*e^3*n^\ 2-1229346*b^2*d*e^3*n^2-1171333*a*b^2*d*e^3*n^2-385326*a^2*b^2*d*e^3*n^2-51494*a\ ^3*b^2*d*e^3*n^2-2484*a^4*b^2*d*e^3*n^2-25*a^5*b^2*d*e^3*n^2-328656*b^3*d*e^3*n^\ 2-288360*a*b^3*d*e^3*n^2-84184*a^2*b^3*d*e^3*n^2-9270*a^3*b^3*d*e^3*n^2-298*a^4*\ b^3*d*e^3*n^2-38808*b^4*d*e^3*n^2-29978*a*b^4*d*e^3*n^2-7029*a^2*b^4*d*e^3*n^2-4\ 89*a^3*b^4*d*e^3*n^2-1484*b^5*d*e^3*n^2-900*a*b^5*d*e^3*n^2-124*a^2*b^5*d*e^3*n^\ 2-3608575*c*d*e^3*n^2-3318732*a*c*d*e^3*n^2-1034145*a^2*c*d*e^3*n^2-126576*a^3*c\ *d*e^3*n^2-5178*a^4*c*d*e^3*n^2-36*a^5*c*d*e^3*n^2-4610124*b*c*d*e^3*n^2-3900978\ *a*b*c*d*e^3*n^2-1109772*a^2*b*c*d*e^3*n^2-122656*a^3*b*c*d*e^3*n^2-4536*a^4*b*c\ *d*e^3*n^2-32*a^5*b*c*d*e^3*n^2-2157301*b^2*c*d*e^3*n^2-1645794*a*b^2*c*d*e^3*n^\ 2-407174*a^2*b^2*c*d*e^3*n^2-36252*a^3*b^2*c*d*e^3*n^2-871*a^4*b^2*c*d*e^3*n^2-4\ 43592*b^3*c*d*e^3*n^2-292304*a*b^3*c*d*e^3*n^2-57042*a^2*b^3*c*d*e^3*n^2-3136*a^\ 3*b^3*c*d*e^3*n^2-37398*b^4*c*d*e^3*n^2-19332*a*b^4*c*d*e^3*n^2-2269*a^2*b^4*c*d\ *e^3*n^2-900*b^5*c*d*e^3*n^2-280*a*b^5*c*d*e^3*n^2-3938634*c^2*d*e^3*n^2-2999785\ *a*c^2*d*e^3*n^2-736677*a^2*c^2*d*e^3*n^2-64236*a^3*c^2*d*e^3*n^2-1431*a^4*c^2*d\ *e^3*n^2-4092639*b*c^2*d*e^3*n^2-2765628*a*b*c^2*d*e^3*n^2-585956*a^2*b*c^2*d*e^\ 3*n^2-41832*a^3*b*c^2*d*e^3*n^2-696*a^4*b*c^2*d*e^3*n^2-1483254*b^2*c^2*d*e^3*n^\ 2-851666*a*b^2*c^2*d*e^3*n^2-140580*a^2*b^2*c^2*d*e^3*n^2-6144*a^3*b^2*c^2*d*e^3\ *n^2-218936*b^3*c^2*d*e^3*n^2-97254*a*b^3*c^2*d*e^3*n^2-9592*a^2*b^3*c^2*d*e^3*n\ ^2-11529*b^4*c^2*d*e^3*n^2-3049*a*b^4*c^2*d*e^3*n^2-124*b^5*c^2*d*e^3*n^2-235070\ 3*c^3*d*e^3*n^2-1421676*a*c^3*d*e^3*n^2-256040*a^2*c^3*d*e^3*n^2-13608*a^3*c^3*d\ *e^3*n^2-84*a^4*c^3*d*e^3*n^2-1904058*b*c^3*d*e^3*n^2-966000*a*b*c^3*d*e^3*n^2-1\ 35288*a^2*b*c^3*d*e^3*n^2-4576*a^3*b*c^3*d*e^3*n^2-498754*b^2*c^3*d*e^3*n^2-1932\ 12*a*b^2*c^3*d*e^3*n^2-16072*a^2*b^2*c^3*d*e^3*n^2-46386*b^3*c^3*d*e^3*n^2-10576\ *a*b^3*c^3*d*e^3*n^2-1109*b^4*c^3*d*e^3*n^2-830619*c^4*d*e^3*n^2-375454*a*c^4*d*\ e^3*n^2-44100*a^2*c^4*d*e^3*n^2-1040*a^3*c^4*d*e^3*n^2-489256*b*c^4*d*e^3*n^2-16\ 7058*a*b*c^4*d*e^3*n^2-11672*a^2*b*c^4*d*e^3*n^2-81468*b^2*c^4*d*e^3*n^2-16211*a\ *b^2*c^4*d*e^3*n^2-3494*b^3*c^4*d*e^3*n^2-173590*c^5*d*e^3*n^2-52614*a*c^5*d*e^3\ *n^2-3040*a^2*c^5*d*e^3*n^2-65484*b*c^5*d*e^3*n^2-11472*a*b*c^5*d*e^3*n^2-5093*b\ ^2*c^5*d*e^3*n^2-19773*c^6*d*e^3*n^2-3068*a*c^6*d*e^3*n^2-3520*b*c^6*d*e^3*n^2-9\ 36*c^7*d*e^3*n^2+637121*d^2*e^3*n^2+557856*a*d^2*e^3*n^2+166035*a^2*d^2*e^3*n^2+\ 20313*a^3*d^2*e^3*n^2+1023*a^4*d^2*e^3*n^2+18*a^5*d^2*e^3*n^2+815040*b*d^2*e^3*n\ ^2+692062*a*b*d^2*e^3*n^2+199602*a^2*b*d^2*e^3*n^2+23350*a^3*b*d^2*e^3*n^2+1035*\ a^4*b*d^2*e^3*n^2+11*a^5*b*d^2*e^3*n^2+370290*b^2*d^2*e^3*n^2+294651*a*b^2*d^2*e\ ^3*n^2+76580*a^2*b^2*d^2*e^3*n^2+7452*a^3*b^2*d^2*e^3*n^2+217*a^4*b^2*d^2*e^3*n^\ 2+70884*b^3*d^2*e^3*n^2+50328*a*b^3*d^2*e^3*n^2+10584*a^2*b^3*d^2*e^3*n^2+648*a^\ 3*b^3*d^2*e^3*n^2+5062*b^4*d^2*e^3*n^2+2889*a*b^4*d^2*e^3*n^2+369*a^2*b^4*d^2*e^\ 3*n^2+72*b^5*d^2*e^3*n^2+24*a*b^5*d^2*e^3*n^2+1372896*c*d^2*e^3*n^2+962202*a*c*d\ ^2*e^3*n^2+211833*a^2*c*d^2*e^3*n^2+16424*a^3*c*d^2*e^3*n^2+414*a^4*c*d^2*e^3*n^\ 2+4*a^5*c*d^2*e^3*n^2+1432642*b*c*d^2*e^3*n^2+944406*a*b*c*d^2*e^3*n^2+193874*a^\ 2*b*c*d^2*e^3*n^2+13716*a^3*b*c*d^2*e^3*n^2+263*a^4*b*c*d^2*e^3*n^2+507303*b^2*c\ *d^2*e^3*n^2+296724*a*b^2*c*d^2*e^3*n^2+49536*a^2*b^2*c*d^2*e^3*n^2+2248*a^3*b^2\ *c*d^2*e^3*n^2+70576*b^3*c*d^2*e^3*n^2+33156*a*b^3*c*d^2*e^3*n^2+3404*a^2*b^3*c*\ d^2*e^3*n^2+3249*b^4*c*d^2*e^3*n^2+942*a*b^4*c*d^2*e^3*n^2+24*b^5*c*d^2*e^3*n^2+\ 1228387*c^2*d^2*e^3*n^2+661986*a*c^2*d^2*e^3*n^2+98628*a^2*c^2*d^2*e^3*n^2+3564*\ a^3*c^2*d^2*e^3*n^2-3*a^4*c^2*d^2*e^3*n^2+1001556*b*c^2*d^2*e^3*n^2+483178*a*b*c\ ^2*d^2*e^3*n^2+62136*a^2*b*c^2*d^2*e^3*n^2+1848*a^3*b*c^2*d^2*e^3*n^2+257936*b^2\ *c^2*d^2*e^3*n^2+99918*a*b^2*c^2*d^2*e^3*n^2+8080*a^2*b^2*c^2*d^2*e^3*n^2+22860*\ b^3*c^2*d^2*e^3*n^2+5456*a*b^3*c^2*d^2*e^3*n^2+489*b^4*c^2*d^2*e^3*n^2+585450*c^\ 3*d^2*e^3*n^2+229856*a*c^3*d^2*e^3*n^2+20484*a^2*c^3*d^2*e^3*n^2+168*a^3*c^3*d^2\ *e^3*n^2+347742*b*c^3*d^2*e^3*n^2+110736*a*b*c^3*d^2*e^3*n^2+6720*a^2*b*c^3*d^2*\ e^3*n^2+57258*b^2*c^3*d^2*e^3*n^2+11260*a*b^2*c^3*d^2*e^3*n^2+2364*b^3*c^3*d^2*e\ ^3*n^2+156343*c^4*d^2*e^3*n^2+40464*a*c^4*d^2*e^3*n^2+1626*a^2*c^4*d^2*e^3*n^2+5\ 9607*b*c^4*d^2*e^3*n^2+9629*a*b*c^4*d^2*e^3*n^2+4615*b^2*c^4*d^2*e^3*n^2+22050*c\ ^5*d^2*e^3*n^2+2900*a*c^5*d^2*e^3*n^2+3985*b*c^5*d^2*e^3*n^2+1269*c^6*d^2*e^3*n^\ 2-126522*d^3*e^3*n^2-86709*a*d^3*e^3*n^2-20313*a^2*d^3*e^3*n^2-2112*a^3*d^3*e^3*\ n^2-99*a^4*d^3*e^3*n^2-a^5*d^3*e^3*n^2-133638*b*d^3*e^3*n^2-92592*a*b*d^3*e^3*n^\ 2-21106*a^2*b*d^3*e^3*n^2-1854*a^3*b*d^3*e^3*n^2-50*a^4*b*d^3*e^3*n^2-45054*b^2*\ d^3*e^3*n^2-29142*a*b^2*d^3*e^3*n^2-5616*a^2*b^2*d^3*e^3*n^2-320*a^3*b^2*d^3*e^3\ *n^2-5408*b^3*d^3*e^3*n^2-2916*a*b^3*d^3*e^3*n^2-356*a^2*b^3*d^3*e^3*n^2-162*b^4\ *d^3*e^3*n^2-54*a*b^4*d^3*e^3*n^2-220347*c*d^3*e^3*n^2-106074*a*c*d^3*e^3*n^2-14\ 180*a^2*c*d^3*e^3*n^2-612*a^3*c*d^3*e^3*n^2-11*a^4*c*d^3*e^3*n^2-182700*b*c*d^3*\ e^3*n^2-86980*a*b*c*d^3*e^3*n^2-11286*a^2*b*c*d^3*e^3*n^2-384*a^3*b*c*d^3*e^3*n^\ 2-45366*b^2*c*d^3*e^3*n^2-18540*a*b^2*c*d^3*e^3*n^2-1620*a^2*b^2*c*d^3*e^3*n^2-3\ 564*b^3*c*d^3*e^3*n^2-944*a*b^3*c*d^3*e^3*n^2-54*b^4*c*d^3*e^3*n^2-157959*c^2*d^\ 3*e^3*n^2-50912*a*c^2*d^3*e^3*n^2-2970*a^2*c^2*d^3*e^3*n^2+16*a^3*c^2*d^3*e^3*n^\ 2-95614*b*c^2*d^3*e^3*n^2-28386*a*b*c^2*d^3*e^3*n^2-1512*a^2*b*c^2*d^3*e^3*n^2-1\ 5336*b^2*c^2*d^3*e^3*n^2-3060*a*b^2*c^2*d^3*e^3*n^2-572*b^3*c^2*d^3*e^3*n^2-5776\ 8*c^3*d^3*e^3*n^2-11520*a*c^3*d^3*e^3*n^2-168*a^2*c^3*d^3*e^3*n^2-22482*b*c^3*d^\ 3*e^3*n^2-3248*a*b*c^3*d^3*e^3*n^2-1712*b^2*c^3*d^3*e^3*n^2-10647*c^4*d^3*e^3*n^\ 2-1039*a*c^4*d^3*e^3*n^2-1974*b*c^4*d^3*e^3*n^2-781*c^5*d^3*e^3*n^2+9945*d^4*e^3\ *n^2+5031*a*d^4*e^3*n^2+1023*a^2*d^4*e^3*n^2+99*a^3*d^4*e^3*n^2+2*a^4*d^4*e^3*n^\ 2+8550*b*d^4*e^3*n^2+4668*a*b*d^4*e^3*n^2+828*a^2*b*d^4*e^3*n^2+47*a^3*b*d^4*e^3\ *n^2+1962*b^2*d^4*e^3*n^2+981*a*b^2*d^4*e^3*n^2+119*a^2*b^2*d^4*e^3*n^2+108*b^3*\ d^4*e^3*n^2+36*a*b^3*d^4*e^3*n^2+13581*c*d^4*e^3*n^2+3462*a*c*d^4*e^3*n^2+207*a^\ 2*c*d^4*e^3*n^2+8*a^3*c*d^4*e^3*n^2+8592*b*c*d^4*e^3*n^2+2502*a*b*c*d^4*e^3*n^2+\ 149*a^2*b*c*d^4*e^3*n^2+1305*b^2*c*d^4*e^3*n^2+290*a*b^2*c*d^4*e^3*n^2+36*b^3*c*\ d^4*e^3*n^2+7653*c^2*d^4*e^3*n^2+900*a*c^2*d^4*e^3*n^2-19*a^2*c^2*d^4*e^3*n^2+31\ 14*b*c^2*d^4*e^3*n^2+383*a*b*c^2*d^4*e^3*n^2+227*b^2*c^2*d^4*e^3*n^2+2016*c^3*d^\ 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59719*a^2*c*e^4*n^2+17667*a^3*c*e^4*n^2+528*a^4*c*e^4*n^2+836226*b*c*e^4*n^2+687\ 704*a*b*c*e^4*n^2+179910*a^2*b*c*e^4*n^2+16458*a^3*b*c*e^4*n^2+360*a^4*b*c*e^4*n\ ^2+466135*b^2*c*e^4*n^2+338787*a*b^2*c*e^4*n^2+76172*a^2*b^2*c*e^4*n^2+5598*a^3*\ b^2*c*e^4*n^2+80*a^4*b^2*c*e^4*n^2+119025*b^3*c*e^4*n^2+74180*a*b^3*c*e^4*n^2+13\ 347*a^2*b^3*c*e^4*n^2+631*a^3*b^3*c*e^4*n^2+13218*b^4*c*e^4*n^2+6507*a*b^4*c*e^4\ *n^2+738*a^2*b^4*c*e^4*n^2+459*b^5*c*e^4*n^2+138*a*b^5*c*e^4*n^2+665082*c^2*e^4*\ n^2+519192*a*c^2*e^4*n^2+127602*a^2*c^2*e^4*n^2+10818*a^3*c^2*e^4*n^2+216*a^4*c^\ 2*e^4*n^2+794366*b*c^2*e^4*n^2+528237*a*b*c^2*e^4*n^2+105752*a^2*b*c^2*e^4*n^2+6\ 615*a^3*b*c^2*e^4*n^2+72*a^4*b*c^2*e^4*n^2+343161*b^2*c^2*e^4*n^2+188951*a*b^2*c\ ^2*e^4*n^2+28935*a^2*b^2*c^2*e^4*n^2+1095*a^3*b^2*c^2*e^4*n^2+63023*b^3*c^2*e^4*\ n^2+26397*a*b^3*c^2*e^4*n^2+2430*a^2*b^3*c^2*e^4*n^2+4392*b^4*c^2*e^4*n^2+1083*a\ *b^4*c^2*e^4*n^2+69*b^5*c^2*e^4*n^2+420340*c^3*e^4*n^2+261810*a*c^3*e^4*n^2+4817\ 4*a^2*c^3*e^4*n^2+2682*a^3*c^3*e^4*n^2+24*a^4*c^3*e^4*n^2+393579*b*c^3*e^4*n^2+1\ 96526*a*b*c^3*e^4*n^2+26334*a^2*b*c^3*e^4*n^2+816*a^3*b*c^3*e^4*n^2+123844*b^2*c\ ^3*e^4*n^2+45639*a*b^2*c^3*e^4*n^2+3528*a^2*b^2*c^3*e^4*n^2+14544*b^3*c^3*e^4*n^\ 2+3067*a*b^3*c^3*e^4*n^2+474*b^4*c^3*e^4*n^2+156897*c^4*e^4*n^2+72379*a*c^4*e^4*\ n^2+8685*a^2*c^4*e^4*n^2+225*a^3*c^4*e^4*n^2+108177*b*c^4*e^4*n^2+35820*a*b*c^4*\ e^4*n^2+2375*a^2*b*c^4*e^4*n^2+22050*b^2*c^4*e^4*n^2+4073*a*b^2*c^4*e^4*n^2+1236\ *b^3*c^4*e^4*n^2+34880*c^5*e^4*n^2+10530*a*c^5*e^4*n^2+608*a^2*c^5*e^4*n^2+15741\ *b*c^5*e^4*n^2+2587*a*b*c^5*e^4*n^2+1554*b^2*c^5*e^4*n^2+4302*c^6*e^4*n^2+636*a*\ c^6*e^4*n^2+951*b*c^6*e^4*n^2+228*c^7*e^4*n^2-214343*d*e^4*n^2-185391*a*d*e^4*n^\ 2-51131*a^2*d*e^4*n^2-5031*a^3*d*e^4*n^2-132*a^4*d*e^4*n^2-308547*b*d*e^4*n^2-25\ 0987*a*b*d*e^4*n^2-65979*a^2*b*d*e^4*n^2-6384*a^3*b*d*e^4*n^2-180*a^4*b*d*e^4*n^\ 2-164536*b^2*d*e^4*n^2-124191*a*b^2*d*e^4*n^2-29729*a^2*b^2*d*e^4*n^2-2484*a^3*b\ ^2*d*e^4*n^2-50*a^4*b^2*d*e^4*n^2-38808*b^3*d*e^4*n^2-26268*a*b^3*d*e^4*n^2-5256\ *a^2*b^3*d*e^4*n^2-298*a^3*b^3*d*e^4*n^2-3710*b^4*d*e^4*n^2-2061*a*b^4*d*e^4*n^2\ -267*a^2*b^4*d*e^4*n^2-90*b^5*d*e^4*n^2-30*a*b^5*d*e^4*n^2-493938*c*d*e^4*n^2-35\ 0680*a*c*d*e^4*n^2-75294*a^2*c*d*e^4*n^2-5178*a^3*c*d*e^4*n^2-72*a^4*c*d*e^4*n^2\ -580059*b*c*d*e^4*n^2-372069*a*b*c*d*e^4*n^2-72062*a^2*b*c*d*e^4*n^2-4536*a^3*b*\ c*d*e^4*n^2-64*a^4*b*c*d*e^4*n^2-240615*b^2*c*d*e^4*n^2-134128*a*b^2*c*d*e^4*n^2\ -21087*a^2*b^2*c*d*e^4*n^2-871*a^3*b^2*c*d*e^4*n^2-41108*b^3*c*d*e^4*n^2-18234*a\ *b^3*c*d*e^4*n^2-1792*a^2*b^3*c*d*e^4*n^2-2511*b^4*c*d*e^4*n^2-684*a*b^4*c*d*e^4\ *n^2-30*b^5*c*d*e^4*n^2-466654*c^2*d*e^4*n^2-257193*a*c^2*d*e^4*n^2-38770*a^2*c^\ 2*d*e^4*n^2-1431*a^3*c^2*d*e^4*n^2-430785*b*c^2*d*e^4*n^2-202056*a*b*c^2*d*e^4*n\ ^2-24858*a^2*b*c^2*d*e^4*n^2-696*a^3*b*c^2*d*e^4*n^2-130793*b^2*c^2*d*e^4*n^2-47\ 655*a*b^2*c^2*d*e^4*n^2-3614*a^2*b^2*c^2*d*e^4*n^2-14400*b^3*c^2*d*e^4*n^2-3160*\ a*b^3*c^2*d*e^4*n^2-417*b^4*c^2*d*e^4*n^2-234009*c^3*d*e^4*n^2-92990*a*c^3*d*e^4\ *n^2-8406*a^2*c^3*d*e^4*n^2-84*a^3*c^3*d*e^4*n^2-159486*b*c^3*d*e^4*n^2-48420*a*\ b*c^3*d*e^4*n^2-2768*a^2*b*c^3*d*e^4*n^2-31518*b^2*c^3*d*e^4*n^2-5653*a*b^2*c^3*\ d*e^4*n^2-1666*b^3*c^3*d*e^4*n^2-66223*c^4*d*e^4*n^2-16830*a*c^4*d*e^4*n^2-665*a\ ^2*c^4*d*e^4*n^2-29610*b*c^4*d*e^4*n^2-4376*a*b*c^4*d*e^4*n^2-2846*b^2*c^4*d*e^4\ *n^2-10071*c^5*d*e^4*n^2-1229*a*c^5*d*e^4*n^2-2212*b*c^5*d*e^4*n^2-645*c^6*d*e^4\ *n^2+75168*d^2*e^4*n^2+51131*a*d^2*e^4*n^2+11322*a^2*d^2*e^4*n^2+1023*a^3*d^2*e^\ 4*n^2+36*a^4*d^2*e^4*n^2+87292*b*d^2*e^4*n^2+58428*a*b*d^2*e^4*n^2+12753*a^2*b*d\ ^2*e^4*n^2+1035*a^3*b*d^2*e^4*n^2+22*a^4*b*d^2*e^4*n^2+34056*b^2*d^2*e^4*n^2+210\ 98*a*b^2*d^2*e^4*n^2+3951*a^2*b^2*d^2*e^4*n^2+217*a^3*b^2*d^2*e^4*n^2+5062*b^3*d\ ^2*e^4*n^2+2637*a*b^3*d^2*e^4*n^2+325*a^2*b^3*d^2*e^4*n^2+216*b^4*d^2*e^4*n^2+72\ *a*b^4*d^2*e^4*n^2+138423*c*d^2*e^4*n^2+67743*a*c*d^2*e^4*n^2+9190*a^2*c*d^2*e^4\ 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e^5*n^2-30*b^4*c*d*e^5*n^2-19530*c^2*d*e^5*n^2-6850*a*c^2*d*e^5*n^2-441*a^2*c^2*\ d*e^5*n^2-15098*b*c^2*d*e^5*n^2-4158*a*b*c^2*d*e^5*n^2-200*a^2*b*c^2*d*e^5*n^2-3\ 393*b^2*c^2*d*e^5*n^2-569*a*b^2*c^2*d*e^5*n^2-200*b^3*c^2*d*e^5*n^2-7782*c^3*d*e\ ^5*n^2-1620*a*c^3*d*e^5*n^2-28*a^2*c^3*d*e^5*n^2-3996*b*c^3*d*e^5*n^2-496*a*b*c^\ 3*d*e^5*n^2-445*b^2*c^3*d*e^5*n^2-1593*c^4*d*e^5*n^2-145*a*c^4*d*e^5*n^2-410*b*c\ ^4*d*e^5*n^2-135*c^5*d*e^5*n^2+2994*d^2*e^5*n^2+1485*a*d^2*e^5*n^2+253*a^2*d^2*e\ ^5*n^2+18*a^3*d^2*e^5*n^2+2970*b*d^2*e^5*n^2+1510*a*b*d^2*e^5*n^2+252*a^2*b*d^2*\ e^5*n^2+11*a^3*b*d^2*e^5*n^2+874*b^2*d^2*e^5*n^2+405*a*b^2*d^2*e^5*n^2+49*a^2*b^\ 2*d^2*e^5*n^2+72*b^3*d^2*e^5*n^2+24*a*b^3*d^2*e^5*n^2+4455*c*d^2*e^5*n^2+1242*a*\ c*d^2*e^5*n^2+90*a^2*c*d^2*e^5*n^2+4*a^3*c*d^2*e^5*n^2+3258*b*c*d^2*e^5*n^2+882*\ a*b*c*d^2*e^5*n^2+61*a^2*b*c*d^2*e^5*n^2+621*b^2*c*d^2*e^5*n^2+118*a*b^2*c*d^2*e\ ^5*n^2+24*b^3*c*d^2*e^5*n^2+2637*c^2*d^2*e^5*n^2+315*a*c^2*d^2*e^5*n^2-5*a^2*c^2\ 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5-15452424*d*n^5-18370924*a*d*n^5-8118432*a^2*d*n^5-1626168*a^3*d*n^5-142461*a^4\ *d*n^5-4110*a^5*d*n^5-18370924*b*d*n^5-19980072*a*b*d*n^5-7927852*a^2*b*d*n^5-13\ 81140*a^3*b*d*n^5-99452*a^4*b*d*n^5-2106*a^5*b*d*n^5-8118432*b^2*d*n^5-7927852*a\ *b^2*d*n^5-2734938*a^2*b^2*d*n^5-391390*a^3*b^2*d*n^5-20682*a^4*b^2*d*n^5-240*a^\ 5*b^2*d*n^5-1626168*b^3*d*n^5-1381140*a*b^3*d*n^5-391390*a^2*b^3*d*n^5-41112*a^3\ *b^3*d*n^5-1192*a^4*b^3*d*n^5-142461*b^4*d*n^5-99452*a*b^4*d*n^5-20682*a^2*b^4*d\ *n^5-1192*a^3*b^4*d*n^5-4110*b^5*d*n^5-2106*a*b^5*d*n^5-240*a^2*b^5*d*n^5-367418\ 48*c*d*n^5-38088540*a*c*d*n^5-14331030*a^2*c*d*n^5-2356632*a^3*c*d*n^5-159453*a^\ 4*c*d*n^5-3159*a^5*c*d*n^5-36216936*b*c*d*n^5-33467280*a*b*c*d*n^5-10885284*a^2*\ b*c*d*n^5-1464774*a^3*b*c*d*n^5-72684*a^4*b*c*d*n^5-794*a^5*b*c*d*n^5-12806356*b\ ^2*c*d*n^5-10222056*a*b^2*c*d*n^5-2717709*a^2*b^2*c*d*n^5-268002*a^3*b^2*c*d*n^5\ -7342*a^4*b^2*c*d*n^5-1950984*b^3*c*d*n^5-1272288*a*b^3*c*d*n^5-247698*a^2*b^3*c\ 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5-25136*a^3*c^4*d*n^5-2474760*b*c^4*d*n^5-1013040*a*b*c^4*d*n^5-92980*a^2*b*c^4*\ d*n^5-289170*b^2*c^4*d*n^5-65214*a*b^2*c^4*d*n^5-8360*b^3*c^4*d*n^5-989904*c^5*d\ *n^5-390852*a*c^5*d*n^5-34440*a^2*c^5*d*n^5-260064*b*c^5*d*n^5-56616*a*b*c^5*d*n\ ^5-13368*b^2*c^5*d*n^5-86688*c^6*d*n^5-18316*a*c^6*d*n^5-10024*b*c^6*d*n^5-2864*\ c^7*d*n^5+8275738*d^2*n^5+8118432*a*d^2*n^5+2833002*a^2*d^2*n^5+420552*a^3*d^2*n\ ^5+24591*a^4*d^2*n^5+387*a^5*d^2*n^5+8118432*b*d^2*n^5+7140352*a*b*d^2*n^5+21571\ 56*a^2*b*d^2*n^5+261070*a^3*b*d^2*n^5+11043*a^4*b*d^2*n^5+92*a^5*b*d^2*n^5+28330\ 02*b^2*d^2*n^5+2157156*a*b^2*d^2*n^5+530364*a^2*b^2*d^2*n^5+46404*a^3*b^2*d^2*n^\ 5+1048*a^4*b^2*d^2*n^5+420552*b^3*d^2*n^5+261070*a*b^3*d^2*n^5+46404*a^2*b^3*d^2\ *n^5+2176*a^3*b^3*d^2*n^5+24591*b^4*d^2*n^5+11043*a*b^4*d^2*n^5+1048*a^2*b^4*d^2\ *n^5+387*b^5*d^2*n^5+92*a*b^5*d^2*n^5+16236864*c*d^2*n^5+13543530*a*c*d^2*n^5+38\ 66562*a^2*c*d^2*n^5+440787*a^3*c*d^2*n^5+17532*a^4*c*d^2*n^5+138*a^5*c*d^2*n^5+1\ 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^2*c*d*e^2*n^6-115100*a*b^2*c*d*e^2*n^6-27522*b^3*c*d*e^2*n^6-3334044*c^2*d*e^2*\ n^6-1061529*a*c^2*d*e^2*n^6-71422*a^2*c^2*d*e^2*n^6-1221150*b*c^2*d*e^2*n^6-2119\ 48*a*b*c^2*d*e^2*n^6-97129*b^2*c^2*d*e^2*n^6-740880*c^3*d*e^2*n^6-119088*a*c^3*d\ *e^2*n^6-131412*b*c^3*d*e^2*n^6-60228*c^4*d*e^2*n^6+1288854*d^2*e^2*n^6+578938*a\ *d^2*e^2*n^6+72261*a^2*d^2*e^2*n^6+2325*a^3*d^2*e^2*n^6+729908*b*d^2*e^2*n^6+244\ 041*a*b*d^2*e^2*n^6+17433*a^2*b*d^2*e^2*n^6+118776*b^2*d^2*e^2*n^6+22420*a*b^2*d\ ^2*e^2*n^6+5156*b^3*d^2*e^2*n^6+1308846*c*d^2*e^2*n^6+389445*a*c*d^2*e^2*n^6+237\ 66*a^2*c*d^2*e^2*n^6+481593*b*c*d^2*e^2*n^6+80642*a*b*c*d^2*e^2*n^6+37888*b^2*c*\ d^2*e^2*n^6+435078*c^2*d^2*e^2*n^6+64555*a*c^2*d^2*e^2*n^6+77741*b*c^2*d^2*e^2*n\ ^6+47334*c^3*d^2*e^2*n^6-142656*d^3*e^2*n^6-40299*a*d^3*e^2*n^6-2325*a^2*d^3*e^2\ *n^6-52752*b*d^3*e^2*n^6-8762*a*b*d^3*e^2*n^6-4020*b^2*d^3*e^2*n^6-93051*c*d^3*e\ ^2*n^6-12672*a*c*d^3*e^2*n^6-16802*b*c*d^3*e^2*n^6-15107*c^2*d^3*e^2*n^6+5523*d^\ 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*b^2*d*e^4*n^6-35763*c*d*e^4*n^6-4909*a*c*d*e^4*n^6-8076*b*c*d*e^4*n^6-6499*c^2*\ d*e^4*n^6+5523*d^2*e^4*n^6+721*a*d^2*e^4*n^6+1227*b*d^2*e^4*n^6+1948*c*d^2*e^4*n\ ^6-143*d^3*e^4*n^6+3425*e^5*n^6+924*a*e^5*n^6+43*a^2*e^5*n^6+1848*b*e^5*n^6+265*\ a*b*e^5*n^6+234*b^2*e^5*n^6+2772*c*e^5*n^6+382*a*c*e^5*n^6+733*b*c*e^5*n^6+542*c\ ^2*e^5*n^6-924*d*e^5*n^6-117*a*d*e^5*n^6-234*b*d*e^5*n^6-351*c*d*e^5*n^6+43*d^2*\ e^5*n^6+7171728*n^7+7433208*a*n^7+2806630*a^2*n^7+465750*a^3*n^7+32110*a^4*n^7+6\ 60*a^5*n^7+7433208*b*n^7+6828540*a*b*n^7+2221074*a^2*b*n^7+301368*a^3*b*n^7+1530\ 0*a^4*b*n^7+176*a^5*b*n^7+2806630*b^2*n^7+2221074*a*b^2*n^7+591180*a^2*b^2*n^7+5\ 9136*a^3*b^2*n^7+1684*a^4*b^2*n^7+465750*b^3*n^7+301368*a*b^3*n^7+59136*a^2*b^3*\ n^7+3320*a^3*b^3*n^7+32110*b^4*n^7+15300*a*b^4*n^7+1684*a^2*b^4*n^7+660*b^5*n^7+\ 176*a*b^5*n^7+14866416*c*n^7+13049440*a*c*n^7+4030236*a^2*c*n^7+516272*a^3*c*n^7\ +24600*a^4*c*n^7+264*a^5*c*n^7+12441800*b*c*n^7+9296208*a*b*c*n^7+2321652*a^2*b*\ c*n^7+216660*a^3*b*c*n^7+5722*a^4*b*c*n^7+3618324*b^2*c*n^7+2182524*a*b^2*c*n^7+\ 396792*a^2*b^2*c*n^7+20518*a^3*b^2*c*n^7+429808*b^3*c*n^7+188124*a*b^3*c*n^7+189\ 06*a^2*b^3*c*n^7+18600*b^4*c*n^7+4478*a*b^4*c*n^7+176*b^5*c*n^7+12441800*c^2*n^7\ +8884296*a*c^2*n^7+2104428*a^2*c^2*n^7+184932*a^3*c^2*n^7+4564*a^4*c^2*n^7+80604\ 72*b*c^2*n^7+4600236*a*b*c^2*n^7+785160*a^2*b*c^2*n^7+37834*a^3*b*c^2*n^7+168794\ 4*b^2*c^2*n^7+691956*a*b^2*c^2*n^7+64620*a^2*b^2*c^2*n^7+126936*b^3*c^2*n^7+2832\ 2*a*b^3*c^2*n^7+2564*b^4*c^2*n^7+5373648*c^3*n^7+2933984*a*c^3*n^7+474912*a^2*c^\ 3*n^7+21520*a^3*c^3*n^7+2516272*b*c^3*n^7+979128*a*b*c^3*n^7+86016*a^2*b*c^3*n^7\ +334944*b^2*c^3*n^7+70416*a*b^2*c^3*n^7+11816*b^3*c^3*n^7+1258136*c^4*n^7+469296\ *a*c^4*n^7+39144*a^2*c^4*n^7+375480*b*c^4*n^7+75252*a*b*c^4*n^7+23508*b^2*c^4*n^\ 7+150192*c^5*n^7+28944*a*c^5*n^7+21120*b*c^5*n^7+7040*c^6*n^7-7433208*d*n^7-6220\ 900*a*d*n^7-1809162*a^2*d*n^7-214904*a^3*d*n^7-9300*a^4*d*n^7-88*a^5*d*n^7-62209\ 00*b*d*n^7-4442148*a*b*d*n^7-1043232*a^2*b*d*n^7-89736*a^3*b*d*n^7-2124*a^4*b*d*\ n^7-1809162*b^2*d*n^7-1043232*a*b^2*d*n^7-177408*a^2*b^2*d*n^7-8348*a^3*b^2*d*n^\ 7-214904*b^3*d*n^7-89736*a*b^3*d*n^7-8348*a^2*b^3*d*n^7-9300*b^4*d*n^7-2124*a*b^\ 4*d*n^7-88*b^5*d*n^7-12441800*c*d*n^7-8472384*a*c*d*n^7-1887204*a^2*c*d*n^7-1532\ 04*a^3*c*d*n^7-3406*a^4*c*d*n^7-8060472*b*c*d*n^7-4400976*a*b*c*d*n^7-705840*a^2\ *b*c*d*n^7-31236*a^3*b*c*d*n^7-1687944*b^2*c*d*n^7-663036*a*b^2*c*d*n^7-57926*a^\ 2*b^2*c*d*n^7-126936*b^3*c*d*n^7-27136*a*b^3*c*d*n^7-2564*b^4*c*d*n^7-8060472*c^\ 2*d*n^7-4201716*a*c^2*d*n^7-639576*a^2*c^2*d*n^7-26726*a^3*c^2*d*n^7-3774408*b*c\ ^2*d*n^7-1407888*a*b*c^2*d*n^7-116300*a^2*b*c^2*d*n^7-502416*b^2*c^2*d*n^7-10154\ 6*a*b^2*c^2*d*n^7-17724*b^3*c^2*d*n^7-2516272*c^3*d*n^7-898056*a*c^3*d*n^7-70560\ *a^2*c^3*d*n^7-750960*b*c^3*d*n^7-144720*a*b*c^3*d*n^7-47016*b^2*c^3*d*n^7-37548\ 0*c^4*d*n^7-69468*a*c^4*d*n^7-52800*b*c^4*d*n^7-21120*c^5*d*n^7+2806630*d^2*n^7+\ 1809162*a*d^2*n^7+373956*a^2*d^2*n^7+27408*a^3*d^2*n^7+526*a^4*d^2*n^7+1809162*b\ *d^2*n^7+940032*a*b*d^2*n^7+140064*a^2*b*d^2*n^7+5572*a^3*b*d^2*n^7+373956*b^2*d\ ^2*n^7+140064*a*b^2*d^2*n^7+11312*a^2*b^2*d^2*n^7+27408*b^3*d^2*n^7+5572*a*b^3*d\ ^2*n^7+526*b^4*d^2*n^7+3618324*c*d^2*n^7+1784004*a*c*d^2*n^7+251208*a^2*c*d^2*n^\ 7+9410*a^3*c*d^2*n^7+1687944*b*c*d^2*n^7+599268*a*b*c*d^2*n^7+45866*a^2*b*c*d^2*\ n^7+222288*b^2*c*d^2*n^7+42912*a*b^2*c*d^2*n^7+7676*b^3*c*d^2*n^7+1687944*c^2*d^\ 2*n^7+570348*a*c^2*d^2*n^7+41436*a^2*c^2*d^2*n^7+502416*b*c^2*d^2*n^7+92350*a*b*\ c^2*d^2*n^7+31184*b^2*c^2*d^2*n^7+334944*c^3*d^2*n^7+58848*a*c^3*d^2*n^7+47016*b\ *c^3*d^2*n^7+23508*c^4*d^2*n^7-465750*d^3*n^7-214904*a*d^3*n^7-27408*a^2*d^3*n^7\ -884*a^3*d^3*n^7-214904*b*d^3*n^7-72120*a*b*d^3*n^7-5024*a^2*b*d^3*n^7-27408*b^2\ *d^3*n^7-5024*a*b^2*d^3*n^7-884*b^3*d^3*n^7-429808*c*d^3*n^7-135588*a*c*d^3*n^7-\ 8862*a^2*c*d^3*n^7-126936*b*c*d^3*n^7-22040*a*b*c*d^3*n^7-7676*b^2*c*d^3*n^7-126\ 936*c^2*d^3*n^7-20854*a*c^2*d^3*n^7-17724*b*c^2*d^3*n^7-11816*c^3*d^3*n^7+32110*\ d^4*n^7+9300*a*d^4*n^7+526*a^2*d^4*n^7+9300*b*d^4*n^7+1512*a*b*d^4*n^7+526*b^2*d\ ^4*n^7+18600*c*d^4*n^7+2794*a*c*d^4*n^7+2564*b*c*d^4*n^7+2564*c^2*d^4*n^7-660*d^\ 5*n^7-88*a*d^5*n^7-88*b*d^5*n^7-176*c*d^5*n^7+7433208*e*n^7+6220900*a*e*n^7+1809\ 162*a^2*e*n^7+214904*a^3*e*n^7+9300*a^4*e*n^7+88*a^5*e*n^7+6828540*b*e*n^7+48540\ 60*a*b*e*n^7+1139292*a^2*b*e*n^7+98388*a^3*b*e*n^7+2354*a^4*b*e*n^7+2221074*b^2*\ e*n^7+1278420*a*b^2*e*n^7+219384*a^2*b^2*e*n^7+10558*a^3*b^2*e*n^7+301368*b^3*e*\ n^7+126924*a*b^3*e*n^7+12170*a^2*b^3*e*n^7+15300*b^4*e*n^7+3598*a*b^4*e*n^7+176*\ b^5*e*n^7+13049440*c*e*n^7+8884296*a*c*e*n^7+1983264*a^2*c*e*n^7+161856*a^3*c*e*\ n^7+3636*a^4*c*e*n^7+9296208*b*c*e*n^7+5041824*a*b*c*e*n^7+808620*a^2*b*c*e*n^7+\ 36084*a^3*b*c*e*n^7+2182524*b^2*c*e*n^7+851424*a*b^2*c*e*n^7+74942*a^2*b^2*c*e*n\ ^7+188124*b^3*c*e*n^7+40184*a*b^3*c*e*n^7+4478*b^4*c*e*n^7+8884296*c^2*e*n^7+460\ 7376*a*c^2*e*n^7+700380*a^2*c^2*e*n^7+29364*a^3*c^2*e*n^7+4600236*b*c^2*e*n^7+16\ 91928*a*b*c^2*e*n^7+138950*a^2*b*c^2*e*n^7+691956*b^2*c^2*e*n^7+137396*a*b^2*c^2\ *e*n^7+28322*b^3*c^2*e*n^7+2933984*c^3*e*n^7+1030896*a*c^3*e*n^7+80016*a^2*c^3*e\ *n^7+979128*b*c^3*e*n^7+183600*a*b*c^3*e*n^7+70416*b^2*c^3*e*n^7+469296*c^4*e*n^\ 7+84072*a*c^4*e*n^7+75252*b*c^4*e*n^7+28944*c^5*e*n^7-6220900*d*e*n^7-4030236*a*\ d*e*n^7-843972*a^2*d*e*n^7-63468*a^3*d*e*n^7-1282*a^4*d*e*n^7-4442148*b*d*e*n^7-\ 2314512*a*b*d*e*n^7-351024*a^2*b*d*e*n^7-14540*a^3*b*d*e*n^7-1043232*b^2*d*e*n^7\ -393828*a*b^2*d*e*n^7-32882*a^2*b^2*d*e*n^7-89736*b^3*d*e*n^7-18640*a*b^3*d*e*n^\ 7-2124*b^4*d*e*n^7-8472384*c*d*e*n^7-4172928*a*c*d*e*n^7-592140*a^2*c*d*e*n^7-22\ 644*a^3*c*d*e*n^7-4400976*b*c*d*e*n^7-1553472*a*b*c*d*e*n^7-120132*a^2*b*c*d*e*n\ ^7-663036*b^2*c*d*e*n^7-127496*a*b^2*c*d*e*n^7-27136*b^3*c*d*e*n^7-4201716*c^2*d\ *e*n^7-1400760*a*c^2*d*e*n^7-101098*a^2*c^2*d*e*n^7-1407888*b*c^2*d*e*n^7-253440\ *a*b*c^2*d*e*n^7-101546*b^2*c^2*d*e*n^7-898056*c^3*d*e*n^7-152688*a*c^3*d*e*n^7-\ 144720*b*c^3*d*e*n^7-69468*c^4*d*e*n^7+1809162*d^2*e*n^7+843972*a*d^2*e*n^7+1111\ 44*a^2*d^2*e*n^7+3838*a^3*d^2*e*n^7+940032*b*d^2*e*n^7+319140*a*b*d^2*e*n^7+2324\ 2*a^2*b*d^2*e*n^7+140064*b^2*d^2*e*n^7+26196*a*b^2*d^2*e*n^7+5572*b^3*d^2*e*n^7+\ 1784004*c*d^2*e*n^7+560256*a*c*d^2*e*n^7+37090*a^2*c*d^2*e*n^7+599268*b*c*d^2*e*\ n^7+103376*a*b*c*d^2*e*n^7+42912*b^2*c*d^2*e*n^7+570348*c^2*d^2*e*n^7+91028*a*c^\ 2*d^2*e*n^7+92350*b*c^2*d^2*e*n^7+58848*c^3*d^2*e*n^7-214904*d^3*e*n^7-63468*a*d\ ^3*e*n^7-3838*a^2*d^3*e*n^7-72120*b*d^3*e*n^7-11992*a*b*d^3*e*n^7-5024*b^2*d^3*e\ *n^7-135588*c*d^3*e*n^7-20096*a*c*d^3*e*n^7-22040*b*c*d^3*e*n^7-20854*c^2*d^3*e*\ n^7+9300*d^4*e*n^7+1282*a*d^4*e*n^7+1512*b*d^4*e*n^7+2794*c*d^4*e*n^7-88*d^5*e*n\ ^7+2806630*e^2*n^7+1809162*a*e^2*n^7+373956*a^2*e^2*n^7+27408*a^3*e^2*n^7+526*a^\ 4*e^2*n^7+2221074*b*e^2*n^7+1139292*a*b*e^2*n^7+168984*a^2*b*e^2*n^7+6758*a^3*b*\ e^2*n^7+591180*b^2*e^2*n^7+219384*a*b^2*e^2*n^7+18006*a^2*b^2*e^2*n^7+59136*b^3*\ e^2*n^7+12170*a*b^3*e^2*n^7+1684*b^4*e^2*n^7+4030236*c*e^2*n^7+1983264*a*c*e^2*n\ ^7+280128*a^2*c*e^2*n^7+10596*a^3*c*e^2*n^7+2321652*b*c*e^2*n^7+808620*a*b*c*e^2\ *n^7+61588*a^2*b*c*e^2*n^7+396792*b^2*c*e^2*n^7+74942*a*b^2*c*e^2*n^7+18906*b^3*\ c*e^2*n^7+2104428*c^2*e^2*n^7+700380*a*c^2*e^2*n^7+50464*a^2*c^2*e^2*n^7+785160*\ b*c^2*e^2*n^7+138950*a*b*c^2*e^2*n^7+64620*b^2*c^2*e^2*n^7+474912*c^3*e^2*n^7+80\ 016*a*c^3*e^2*n^7+86016*b*c^3*e^2*n^7+39144*c^4*e^2*n^7-1809162*d*e^2*n^7-843972\ *a*d*e^2*n^7-111144*a^2*d*e^2*n^7-3838*a^3*d*e^2*n^7-1043232*b*d*e^2*n^7-351024*\ a*b*d*e^2*n^7-25492*a^2*b*d*e^2*n^7-177408*b^2*d*e^2*n^7-32882*a*b^2*d*e^2*n^7-8\ 348*b^3*d*e^2*n^7-1887204*c*d*e^2*n^7-592140*a*c*d*e^2*n^7-39340*a^2*c*d*e^2*n^7\ -705840*b*c*d*e^2*n^7-120132*a*b*c*d*e^2*n^7-57926*b^2*c*d*e^2*n^7-639576*c^2*d*\ e^2*n^7-101098*a*c^2*d*e^2*n^7-116300*b*c^2*d*e^2*n^7-70560*c^3*d*e^2*n^7+373956\ *d^2*e^2*n^7+111144*a*d^2*e^2*n^7+6882*a^2*d^2*e^2*n^7+140064*b*d^2*e^2*n^7+2324\ 2*a*b*d^2*e^2*n^7+11312*b^2*d^2*e^2*n^7+251208*c*d^2*e^2*n^7+37090*a*c*d^2*e^2*n\ ^7+45866*b*c*d^2*e^2*n^7+41436*c^2*d^2*e^2*n^7-27408*d^3*e^2*n^7-3838*a*d^3*e^2*\ n^7-5024*b*d^3*e^2*n^7-8862*c*d^3*e^2*n^7+526*d^4*e^2*n^7+465750*e^3*n^7+214904*\ a*e^3*n^7+27408*a^2*e^3*n^7+884*a^3*e^3*n^7+301368*b*e^3*n^7+98388*a*b*e^3*n^7+6\ 758*a^2*b*e^3*n^7+59136*b^2*e^3*n^7+10558*a*b^2*e^3*n^7+3320*b^3*e^3*n^7+516272*\ c*e^3*n^7+161856*a*c*e^3*n^7+10596*a^2*c*e^3*n^7+216660*b*c*e^3*n^7+36084*a*b*c*\ e^3*n^7+20518*b^2*c*e^3*n^7+184932*c^2*e^3*n^7+29364*a*c^2*e^3*n^7+37834*b*c^2*e\ ^3*n^7+21520*c^3*e^3*n^7-214904*d*e^3*n^7-63468*a*d*e^3*n^7-3838*a^2*d*e^3*n^7-8\ 9736*b*d*e^3*n^7-14540*a*b*d*e^3*n^7-8348*b^2*d*e^3*n^7-153204*c*d*e^3*n^7-22644\ *a*c*d*e^3*n^7-31236*b*c*d*e^3*n^7-26726*c^2*d*e^3*n^7+27408*d^2*e^3*n^7+3838*a*\ d^2*e^3*n^7+5572*b*d^2*e^3*n^7+9410*c*d^2*e^3*n^7-884*d^3*e^3*n^7+32110*e^4*n^7+\ 9300*a*e^4*n^7+526*a^2*e^4*n^7+15300*b*e^4*n^7+2354*a*b*e^4*n^7+1684*b^2*e^4*n^7\ +24600*c*e^4*n^7+3636*a*c*e^4*n^7+5722*b*c*e^4*n^7+4564*c^2*e^4*n^7-9300*d*e^4*n\ ^7-1282*a*d*e^4*n^7-2124*b*d*e^4*n^7-3406*c*d*e^4*n^7+526*d^2*e^4*n^7+660*e^5*n^\ 7+88*a*e^5*n^7+176*b*e^5*n^7+264*c*e^5*n^7-88*d*e^5*n^7+3716604*n^8+3186405*a*n^\ 8+956070*a^2*n^8+118260*a^3*n^8+5400*a^4*n^8+55*a^5*n^8+3186405*b*n^8+2324052*a*\ b*n^8+563022*a^2*b*n^8+50598*a^3*b*n^8+1275*a^4*b*n^8+956070*b^2*n^8+563022*a*b^\ 2*n^8+99198*a^2*b^2*n^8+4928*a^3*b^2*n^8+118260*b^3*n^8+50598*a*b^3*n^8+4928*a^2\ *b^3*n^8+5400*b^4*n^8+1275*a*b^4*n^8+55*b^5*n^8+6372810*c*n^8+4442148*a*c*n^8+10\ 21923*a^2*c*n^8+86697*a^3*c*n^8+2050*a^4*c*n^8+4236192*b*c*n^8+2356209*a*b*c*n^8\ +389583*a^2*b*c*n^8+18055*a^3*b*c*n^8+917802*b^2*c*n^8+366282*a*b^2*c*n^8+33066*\ a^2*b^2*c*n^8+72198*b^3*c*n^8+15677*a*b^3*c*n^8+1550*b^4*c*n^8+4236192*c^2*n^8+2\ 252088*a*c^2*n^8+353187*a^2*c^2*n^8+15411*a^3*c^2*n^8+2043846*b*c^2*n^8+771957*a\ *b*c^2*n^8+65430*a^2*b*c^2*n^8+283392*b^2*c^2*n^8+57663*a*b^2*c^2*n^8+10578*b^3*\ c^2*n^8+1362564*c^3*n^8+492372*a*c^3*n^8+39576*a^2*c^3*n^8+422388*b*c^3*n^8+8159\ 4*a*b*c^3*n^8+27912*b^2*c^3*n^8+211194*c^4*n^8+39108*a*c^4*n^8+31290*b*c^4*n^8+1\ 2516*c^5*n^8-3186405*d*n^8-2118096*a*d*n^8-458901*a^2*d*n^8-36099*a^3*d*n^8-775*\ a^4*d*n^8-2118096*b*d*n^8-1126044*a*b*d*n^8-175095*a^2*b*d*n^8-7478*a^3*b*d*n^8-\ 458901*b^2*d*n^8-175095*a*b^2*d*n^8-14784*a^2*b^2*d*n^8-36099*b^3*d*n^8-7478*a*b\ ^3*d*n^8-775*b^4*d*n^8-4236192*c*d*n^8-2147967*a*c*d*n^8-316791*a^2*c*d*n^8-1276\ 7*a^3*c*d*n^8-2043846*b*c*d*n^8-738558*a*b*c*d*n^8-58820*a^2*b*c*d*n^8-283392*b^\ 2*c*d*n^8-55253*a*b^2*c*d*n^8-10578*b^3*c*d*n^8-2043846*c^2*d*n^8-705159*a*c^2*d\ *n^8-53298*a^2*c^2*d*n^8-633582*b*c^2*d*n^8-117324*a*b*c^2*d*n^8-41868*b^2*c^2*d\ *n^8-422388*c^3*d*n^8-74838*a*c^3*d*n^8-62580*b*c^3*d*n^8-31290*c^4*d*n^8+956070\ *d^2*n^8+458901*a*d^2*n^8+62802*a^2*d^2*n^8+2284*a^3*d^2*n^8+458901*b*d^2*n^8+15\ 7788*a*b*d^2*n^8+11672*a^2*b*d^2*n^8+62802*b^2*d^2*n^8+11672*a*b^2*d^2*n^8+2284*\ b^3*d^2*n^8+917802*c*d^2*n^8+299484*a*c*d^2*n^8+20934*a^2*c*d^2*n^8+283392*b*c*d\ ^2*n^8+49939*a*b*c*d^2*n^8+18524*b^2*c*d^2*n^8+283392*c^2*d^2*n^8+47529*a*c^2*d^\ 2*n^8+41868*b*c^2*d^2*n^8+27912*c^3*d^2*n^8-118260*d^3*n^8-36099*a*d^3*n^8-2284*\ a^2*d^3*n^8-36099*b*d^3*n^8-6010*a*b*d^3*n^8-2284*b^2*d^3*n^8-72198*c*d^3*n^8-11\ 299*a*c*d^3*n^8-10578*b*c*d^3*n^8-10578*c^2*d^3*n^8+5400*d^4*n^8+775*a*d^4*n^8+7\ 75*b*d^4*n^8+1550*c*d^4*n^8-55*d^5*n^8+3186405*e*n^8+2118096*a*e*n^8+458901*a^2*\ e*n^8+36099*a^3*e*n^8+775*a^4*e*n^8+2324052*b*e*n^8+1230165*a*b*e*n^8+191187*a^2\ *b*e*n^8+8199*a^3*b*e*n^8+563022*b^2*e*n^8+214488*a*b^2*e*n^8+18282*a^2*b^2*e*n^\ 8+50598*b^3*e*n^8+10577*a*b^3*e*n^8+1275*b^4*e*n^8+4442148*c*e*n^8+2252088*a*c*e\ *n^8+332883*a^2*c*e*n^8+13488*a^3*c*e*n^8+2356209*b*c*e*n^8+845964*a*b*c*e*n^8+6\ 7385*a^2*b*c*e*n^8+366282*b^2*c*e*n^8+70952*a*b^2*c*e*n^8+15677*b^3*c*e*n^8+2252\ 088*c^2*e*n^8+773172*a*c^2*e*n^8+58365*a^2*c^2*e*n^8+771957*b*c^2*e*n^8+140994*a\ *b*c^2*e*n^8+57663*b^2*c^2*e*n^8+492372*c^3*e*n^8+85908*a*c^3*e*n^8+81594*b*c^3*\ e*n^8+39108*c^4*e*n^8-2118096*d*e*n^8-1021923*a*d*e*n^8-141696*a^2*d*e*n^8-5289*\ a^3*d*e*n^8-1126044*b*d*e*n^8-388368*a*b*d*e*n^8-29252*a^2*b*d*e*n^8-175095*b^2*\ d*e*n^8-32819*a*b^2*d*e*n^8-7478*b^3*d*e*n^8-2147967*c*d*e*n^8-700380*a*c*d*e*n^\ 8-49345*a^2*c*d*e*n^8-738558*b*c*d*e*n^8-129456*a*b*c*d*e*n^8-55253*b^2*c*d*e*n^\ 8-705159*c^2*d*e*n^8-116730*a*c^2*d*e*n^8-117324*b*c^2*d*e*n^8-74838*c^3*d*e*n^8\ +458901*d^2*e*n^8+141696*a*d^2*e*n^8+9262*a^2*d^2*e*n^8+157788*b*d^2*e*n^8+26595\ *a*b*d^2*e*n^8+11672*b^2*d^2*e*n^8+299484*c*d^2*e*n^8+46688*a*c*d^2*e*n^8+49939*\ b*c*d^2*e*n^8+47529*c^2*d^2*e*n^8-36099*d^3*e*n^8-5289*a*d^3*e*n^8-6010*b*d^3*e*\ n^8-11299*c*d^3*e*n^8+775*d^4*e*n^8+956070*e^2*n^8+458901*a*e^2*n^8+62802*a^2*e^\ 2*n^8+2284*a^3*e^2*n^8+563022*b*e^2*n^8+191187*a*b*e^2*n^8+14082*a^2*b*e^2*n^8+9\ 9198*b^2*e^2*n^8+18282*a*b^2*e^2*n^8+4928*b^3*e^2*n^8+1021923*c*e^2*n^8+332883*a\ *c*e^2*n^8+23344*a^2*c*e^2*n^8+389583*b*c*e^2*n^8+67385*a*b*c*e^2*n^8+33066*b^2*\ c*e^2*n^8+353187*c^2*e^2*n^8+58365*a*c^2*e^2*n^8+65430*b*c^2*e^2*n^8+39576*c^3*e\ ^2*n^8-458901*d*e^2*n^8-141696*a*d*e^2*n^8-9262*a^2*d*e^2*n^8-175095*b*d*e^2*n^8\ -29252*a*b*d*e^2*n^8-14784*b^2*d*e^2*n^8-316791*c*d*e^2*n^8-49345*a*c*d*e^2*n^8-\ 58820*b*c*d*e^2*n^8-53298*c^2*d*e^2*n^8+62802*d^2*e^2*n^8+9262*a*d^2*e^2*n^8+116\ 72*b*d^2*e^2*n^8+20934*c*d^2*e^2*n^8-2284*d^3*e^2*n^8+118260*e^3*n^8+36099*a*e^3\ *n^8+2284*a^2*e^3*n^8+50598*b*e^3*n^8+8199*a*b*e^3*n^8+4928*b^2*e^3*n^8+86697*c*\ e^3*n^8+13488*a*c*e^3*n^8+18055*b*c*e^3*n^8+15411*c^2*e^3*n^8-36099*d*e^3*n^8-52\ 89*a*d*e^3*n^8-7478*b*d*e^3*n^8-12767*c*d*e^3*n^8+2284*d^2*e^3*n^8+5400*e^4*n^8+\ 775*a*e^4*n^8+1275*b*e^4*n^8+2050*c*e^4*n^8-775*d*e^4*n^8+55*e^5*n^8+1416180*n^9\ +964260*a*n^9+215525*a^2*n^9+17655*a^3*n^9+400*a^4*n^9+964260*b*n^9+523602*a*b*n\ ^9+83985*a^2*b*n^9+3748*a^3*b*n^9+215525*b^2*n^9+83985*a*b^2*n^9+7348*a^2*b^2*n^\ 9+17655*b^3*n^9+3748*a*b^3*n^9+400*b^4*n^9+1928520*c*n^9+1000928*a*c*n^9+152460*\ a^2*c*n^9+6422*a^3*c*n^9+954652*b*c*n^9+351450*a*b*c*n^9+28858*a^2*b*c*n^9+13695\ 0*b^2*c*n^9+27132*a*b^2*c*n^9+5348*b^3*c*n^9+954652*c^2*n^9+335940*a*c^2*n^9+261\ 62*a^2*c^2*n^9+304920*b*c^2*n^9+57182*a*b*c^2*n^9+20992*b^2*c^2*n^9+203280*c^3*n\ ^9+36472*a*c^3*n^9+31288*b*c^3*n^9+15644*c^4*n^9-964260*d*n^9-477326*a*d*n^9-684\ 75*a^2*d*n^9-2674*a^3*d*n^9-477326*b*d*n^9-167970*a*b*d*n^9-12970*a^2*b*d*n^9-68\ 475*b^2*d*n^9-12970*a*b^2*d*n^9-2674*b^3*d*n^9-954652*c*d*n^9-320430*a*c*d*n^9-2\ 3466*a^2*c*d*n^9-304920*b*c*d*n^9-54708*a*b*c*d*n^9-20992*b^2*c*d*n^9-304920*c^2\ *d*n^9-52234*a*c^2*d*n^9-46932*b*c^2*d*n^9-31288*c^3*d*n^9+215525*d^2*n^9+68475*\ a*d^2*n^9+4652*a^2*d^2*n^9+68475*b*d^2*n^9+11688*a*b*d^2*n^9+4652*b^2*d^2*n^9+13\ 6950*c*d^2*n^9+22184*a*c*d^2*n^9+20992*b*c*d^2*n^9+20992*c^2*d^2*n^9-17655*d^3*n\ ^9-2674*a*d^3*n^9-2674*b*d^3*n^9-5348*c*d^3*n^9+400*d^4*n^9+964260*e*n^9+477326*\ a*e*n^9+68475*a^2*e*n^9+2674*a^3*e*n^9+523602*b*e*n^9+183480*a*b*e*n^9+14162*a^2\ *b*e*n^9+83985*b^2*e*n^9+15888*a*b^2*e*n^9+3748*b^3*e*n^9+1000928*c*e*n^9+335940\ *a*c*e*n^9+24658*a^2*c*e*n^9+351450*b*c*e*n^9+62664*a*b*c*e*n^9+27132*b^2*c*e*n^\ 9+335940*c^2*e*n^9+57272*a*c^2*e*n^9+57182*b*c^2*e*n^9+36472*c^3*e*n^9-477326*d*\ e*n^9-152460*a*d*e*n^9-10496*a^2*d*e*n^9-167970*b*d*e*n^9-28768*a*b*d*e*n^9-1297\ 0*b^2*d*e*n^9-320430*c*d*e*n^9-51880*a*c*d*e*n^9-54708*b*c*d*e*n^9-52234*c^2*d*e\ *n^9+68475*d^2*e*n^9+10496*a*d^2*e*n^9+11688*b*d^2*e*n^9+22184*c*d^2*e*n^9-2674*\ d^3*e*n^9+215525*e^2*n^9+68475*a*e^2*n^9+4652*a^2*e^2*n^9+83985*b*e^2*n^9+14162*\ a*b*e^2*n^9+7348*b^2*e^2*n^9+152460*c*e^2*n^9+24658*a*c*e^2*n^9+28858*b*c*e^2*n^\ 9+26162*c^2*e^2*n^9-68475*d*e^2*n^9-10496*a*d*e^2*n^9-12970*b*d*e^2*n^9-23466*c*\ d*e^2*n^9+4652*d^2*e^2*n^9+17655*e^3*n^9+2674*a*e^3*n^9+3748*b*e^3*n^9+6422*c*e^\ 3*n^9-2674*d*e^3*n^9+400*e^4*n^9+385704*n^10+195558*a*n^10+28941*a^2*n^10+1177*a\ ^3*n^10+195558*b*n^10+70290*a*b*n^10+5599*a^2*b*n^10+28941*b^2*n^10+5599*a*b^2*n\ ^10+1177*b^3*n^10+391116*c*n^10+134376*a*c*n^10+10164*a^2*c*n^10+128172*b*c*n^10\ +23430*a*b*c*n^10+9130*b^2*c*n^10+128172*c^2*n^10+22396*a*c^2*n^10+20328*b*c^2*n\ ^10+13552*c^3*n^10-195558*d*n^10-64086*a*d*n^10-4565*a^2*d*n^10-64086*b*d*n^10-1\ 1198*a*b*d*n^10-4565*b^2*d*n^10-128172*c*d*n^10-21362*a*c*d*n^10-20328*b*c*d*n^1\ 0-20328*c^2*d*n^10+28941*d^2*n^10+4565*a*d^2*n^10+4565*b*d^2*n^10+9130*c*d^2*n^1\ 0-1177*d^3*n^10+195558*e*n^10+64086*a*e*n^10+4565*a^2*e*n^10+70290*b*e*n^10+1223\ 2*a*b*e*n^10+5599*b^2*e*n^10+134376*c*e*n^10+22396*a*c*e*n^10+23430*b*c*e*n^10+2\ 2396*c^2*e*n^10-64086*d*e*n^10-10164*a*d*e*n^10-11198*b*d*e*n^10-21362*c*d*e*n^1\ 0+4565*d^2*e*n^10+28941*e^2*n^10+4565*a*e^2*n^10+5599*b*e^2*n^10+10164*c*e^2*n^1\ 0-4565*d*e^2*n^10+1177*e^3*n^10+71112*n^11+23868*a*n^11+1754*a^2*n^11+23868*b*n^\ 11+4260*a*b*n^11+1754*b^2*n^11+47736*c*n^11+8144*a*c*n^11+7768*b*c*n^11+7768*c^2\ *n^11-23868*d*n^11-3884*a*d*n^11-3884*b*d*n^11-7768*c*d*n^11+1754*d^2*n^11+23868\ *e*n^11+3884*a*e*n^11+4260*b*e*n^11+8144*c*e*n^11-3884*d*e*n^11+1754*e^2*n^11+79\ 56*n^12+1326*a*n^12+1326*b*n^12+2652*c*n^12-1326*d*n^12+1326*e*n^12+408*n^13)*Sn\ -((1+a+n)^2*(1+b+n)*(1+c+n)^2*(1+b+c-d+n)*(1+e+n)*(1+c-d+e+n)*(384+240*a+40*a^2+\ 240*b+144*a*b+24*a^2*b+40*b^2+24*a*b^2+4*a^2*b^2+480*c+256*a*c+36*a^2*c+224*b*c+\ 108*a*b*c+14*a^2*b*c+24*b^2*c+10*a*b^2*c+a^2*b^2*c+224*c^2+96*a*c^2+10*a^2*c^2+7\ 2*b*c^2+26*a*b*c^2+2*a^2*b*c^2+4*b^2*c^2+a*b^2*c^2+48*c^3+16*a*c^3+a^2*c^3+8*b*c\ ^3+2*a*b*c^3+4*c^4+a*c^4-240*d-112*a*d-12*a^2*d-112*b*d-48*a*b*d-4*a^2*b*d-12*b^\ 2*d-4*a*b^2*d-224*c*d-84*a*c*d-6*a^2*c*d-72*b*c*d-24*a*b*c*d-a^2*b*c*d-4*b^2*c*d\ -a*b^2*c*d-72*c^2*d-22*a*c^2*d-a^2*c^2*d-12*b*c^2*d-3*a*b*c^2*d-8*c^3*d-2*a*c^3*\ d+40*d^2+12*a*d^2+12*b*d^2+4*a*b*d^2+24*c*d^2+6*a*c*d^2+4*b*c*d^2+a*b*c*d^2+4*c^\ 2*d^2+a*c^2*d^2+240*e+112*a*e+12*a^2*e+144*b*e+60*a*b*e+6*a^2*b*e+24*b^2*e+10*a*\ b^2*e+a^2*b^2*e+256*c*e+96*a*c*e+8*a^2*c*e+108*b*c*e+32*a*b*c*e+2*a^2*b*c*e+10*b\ ^2*c*e+2*a*b^2*c*e+96*c^2*e+24*a*c^2*e+a^2*c^2*e+26*b*c^2*e+4*a*b*c^2*e+b^2*c^2*\ e+16*c^3*e+2*a*c^3*e+2*b*c^3*e+c^4*e-112*d*e-36*a*d*e-2*a^2*d*e-48*b*d*e-16*a*b*\ d*e-a^2*b*d*e-4*b^2*d*e-a*b^2*d*e-84*c*d*e-16*a*c*d*e-24*b*c*d*e-4*a*b*c*d*e-b^2\ *c*d*e-22*c^2*d*e-2*a*c^2*d*e-3*b*c^2*d*e-2*c^3*d*e+12*d^2*e+2*a*d^2*e+4*b*d^2*e\ +a*b*d^2*e+6*c*d^2*e+b*c*d^2*e+c^2*d^2*e+40*e^2+12*a*e^2+24*b*e^2+6*a*b*e^2+4*b^\ 2*e^2+a*b^2*e^2+36*c*e^2+8*a*c*e^2+14*b*c*e^2+2*a*b*c*e^2+b^2*c*e^2+10*c^2*e^2+a\ *c^2*e^2+2*b*c^2*e^2+c^3*e^2-12*d*e^2-2*a*d*e^2-4*b*d*e^2-a*b*d*e^2-6*c*d*e^2-b*\ c*d*e^2-c^2*d*e^2+960*n+480*a*n+60*a^2*n+480*b*n+216*a*b*n+24*a^2*b*n+60*b^2*n+2\ 4*a*b^2*n+2*a^2*b^2*n+960*c*n+384*a*c*n+36*a^2*c*n+336*b*c*n+108*a*b*c*n+7*a^2*b\ *c*n+24*b^2*c*n+5*a*b^2*c*n+336*c^2*n+96*a*c^2*n+5*a^2*c^2*n+72*b*c^2*n+13*a*b*c\ ^2*n+2*b^2*c^2*n+48*c^3*n+8*a*c^3*n+4*b*c^3*n+2*c^4*n-480*d*n-168*a*d*n-12*a^2*d\ *n-168*b*d*n-48*a*b*d*n-2*a^2*b*d*n-12*b^2*d*n-2*a*b^2*d*n-336*c*d*n-84*a*c*d*n-\ 3*a^2*c*d*n-72*b*c*d*n-12*a*b*c*d*n-2*b^2*c*d*n-72*c^2*d*n-11*a*c^2*d*n-6*b*c^2*\ d*n-4*c^3*d*n+60*d^2*n+12*a*d^2*n+12*b*d^2*n+2*a*b*d^2*n+24*c*d^2*n+3*a*c*d^2*n+\ 2*b*c*d^2*n+2*c^2*d^2*n+480*e*n+168*a*e*n+12*a^2*e*n+216*b*e*n+60*a*b*e*n+3*a^2*\ b*e*n+24*b^2*e*n+5*a*b^2*e*n+384*c*e*n+96*a*c*e*n+4*a^2*c*e*n+108*b*c*e*n+16*a*b\ *c*e*n+5*b^2*c*e*n+96*c^2*e*n+12*a*c^2*e*n+13*b*c^2*e*n+8*c^3*e*n-168*d*e*n-36*a\ *d*e*n-a^2*d*e*n-48*b*d*e*n-8*a*b*d*e*n-2*b^2*d*e*n-84*c*d*e*n-8*a*c*d*e*n-12*b*\ c*d*e*n-11*c^2*d*e*n+12*d^2*e*n+a*d^2*e*n+2*b*d^2*e*n+3*c*d^2*e*n+60*e^2*n+12*a*\ e^2*n+24*b*e^2*n+3*a*b*e^2*n+2*b^2*e^2*n+36*c*e^2*n+4*a*c*e^2*n+7*b*c*e^2*n+5*c^\ 2*e^2*n-12*d*e^2*n-a*d*e^2*n-2*b*d*e^2*n-3*c*d*e^2*n+960*n^2+360*a*n^2+30*a^2*n^\ 2+360*b*n^2+108*a*b*n^2+6*a^2*b*n^2+30*b^2*n^2+6*a*b^2*n^2+720*c*n^2+192*a*c*n^2\ +9*a^2*c*n^2+168*b*c*n^2+27*a*b*c*n^2+6*b^2*c*n^2+168*c^2*n^2+24*a*c^2*n^2+18*b*\ c^2*n^2+12*c^3*n^2-360*d*n^2-84*a*d*n^2-3*a^2*d*n^2-84*b*d*n^2-12*a*b*d*n^2-3*b^\ 2*d*n^2-168*c*d*n^2-21*a*c*d*n^2-18*b*c*d*n^2-18*c^2*d*n^2+30*d^2*n^2+3*a*d^2*n^\ 2+3*b*d^2*n^2+6*c*d^2*n^2+360*e*n^2+84*a*e*n^2+3*a^2*e*n^2+108*b*e*n^2+15*a*b*e*\ n^2+6*b^2*e*n^2+192*c*e*n^2+24*a*c*e*n^2+27*b*c*e*n^2+24*c^2*e*n^2-84*d*e*n^2-9*\ a*d*e*n^2-12*b*d*e*n^2-21*c*d*e*n^2+3*d^2*e*n^2+30*e^2*n^2+3*a*e^2*n^2+6*b*e^2*n\ ^2+9*c*e^2*n^2-3*d*e^2*n^2+480*n^3+120*a*n^3+5*a^2*n^3+120*b*n^3+18*a*b*n^3+5*b^\ 2*n^3+240*c*n^3+32*a*c*n^3+28*b*c*n^3+28*c^2*n^3-120*d*n^3-14*a*d*n^3-14*b*d*n^3\ -28*c*d*n^3+5*d^2*n^3+120*e*n^3+14*a*e*n^3+18*b*e*n^3+32*c*e*n^3-14*d*e*n^3+5*e^\ 2*n^3+120*n^4+15*a*n^4+15*b*n^4+30*c*n^4-15*d*n^4+15*e*n^4+12*n^5)): ope:=NorOp(ope,Sn): [ope,[b, c, e, a, a, c, d]] end: #Search(N,K): All the pentuples [a,b,c,d,e]=[a1/N,a2/N,a3/N,a4/N,a5/N] with the ai's beteen 0 and N-1 such that #the deltas for AnBn(a,b,c,d,e) seems to be positive. Try #Search(2,30); Search:=proc(N,K) local i,gu,gu1, mu,a1,a2,a3,a4,a5,ALD,mu1: ALD:={}: gu:=[]: for a1 from 0 to N-1 do for a2 from 0 to N-1 do for a3 from 0 to N-1 do for a4 from 0 to N-1 do for a5 from 0 to N-1 do gu1:=[a1/N,a2/N,a3/N,a4/N,a5/N]: mu1:= [gu1[2], gu1[3], gu1[5], gu1[1], gu1[1], gu1[3], gu1[4]]: if not member(mu1,ALD) then if lcm(seq(denom(gu1[i]),i=1..nops(gu1)))=N then mu:=CnDn(op(gu1),K): if mu<>FAIL then if min(op(mu[3]))>0 then gu:=[op(gu), gu1]: ALD:=ALD union {mu1}: fi: fi: fi: fi: od: od: od: od: od: gu: end: #Searchold(N,K): All the pentuples [a,b,c,d,e]=[a1/N,a2/N,a3/N,a4/N,a5/N] with the ai's beteen 0 and N-1 such that #deltSeqZ3(a,b,c,d,e,K) seems to be positive. Try #Search(2,30); Searchold:=proc(N,K) local gu,gu1, mu,a1,a2,a3,a4,a5: gu:=[]: for a1 from 0 to N-1 do for a2 from 0 to N-1 do for a3 from 0 to N-1 do for a4 from 0 to N-1 do for a5 from 0 to N-1 do gu1:=[a1/N,a2/N,a3/N,a4/N,a5/N]: mu:=deltSeqZ3(op(gu1),K): if mu[1]<>[] and mu[1][-1]>0 then gu:=[op(gu), gu1]: fi: od: od: od: od: od: gu: end: #IsGoodOpe(ope,n,N): Is ope(n,N) a good operator? IsGoodOpe:=proc(ope,n,N) local gu: gu:=denom(ope): if subs(n=0,gu)<>0 then true: else false: fi: end: #An(a,b,c,d,e,n): The numerator An in the n-th term of the rational arroximation to #IntGBZ3(b, c, e, a, a, c, d,n): #Try: #An(0,0,0,0,0,50); An:=proc(a,b,c,d,e,n) :AnBn(a,b,c,d,e,n)[1][-1]:end: #Bn(a,b,c,d,e,n): The denominator Bn in the n-th term of the rational arroximation to #IntGBZ3(b, c, e, a, a, c, d,n): #Try: #Bn(0,0,0,0,0,50); Bn:=proc(a,b,c,d,e,n) :AnBn(a,b,c,d,e,n)[2][-1]:end: #Cn(a,b,c,d,e,n): The denominator, Cn, in the "integerating sequence En" of the rational arroximation to #IntGBZ3(b, c, e, a, a, c, d,n): #Try: #Cn(0,0,0,0,0,50); Cn:=proc(a,b,c,d,e,n) :CnDn(a,b,c,d,e,n)[1][1][-1]:end: #Dn(a,b,c,d,e,n): The numerator, Dn, in the "integerating sequence En" of the rational arroximation to #IntGBZ3(b, c, e, a, a, c, d,n): #Try: #Dn(0,0,0,0,0,50); Dn:=proc(a,b,c,d,e,n) :CnDn(a,b,c,d,e,n)[1][2][-1]:end: #AnBn(a,b,c,d,e,K): The first K terms in the sequences An and Bn in the expression #for IntGBZ3(b, c, e, a, a, c, d,n): #Try: #AnBn(0,0,0,0,0,50); AnBn:=proc(a,b,c,d,e,K) local ope,n,N,RE,c0,c1,L,a0,i,gu: option remember: RE:=FindRelGZ3(b, c, e, a, a, c, d): if RE=FAIL then RETURN(FAIL): fi: c0:=RE[1][1]: c1:=RE[1][2]: if c1=0 then RETURN(FAIL): fi: L:=RE[2]: ope:=OPEZ3(a,b,c,d,e,n,N)[1]: if not IsGoodOpe(ope,n,N) then RETURN(FAIL): fi: gu:=SeqFromRec(ope,n,N,[a0,(L-c0*a0)/c1],K): [[seq(-coeff(gu[i],a0,0),i=1..nops(gu))],[seq(coeff(gu[i],a0,1),i=1..nops(gu))]]: end: #CnDn(a,b,c,d,e,K): The first K terms in the sequences gcd(numer(An[i]),number(Bn[i])) #lcm(denom(An[i]),denom(Bn[i])) where An, Bn are the sequences given by AnBn(a,b,c,d,e,K) (q.v.) #It also returns the last 20 terms of the logs of the normailizing sequence #as well as the last 20 terms of the implied deltas #Try: #CnDn(0,0,0,0,0,50); CnDn:=proc(a,b,c,d,e,K) local gu,i,lu,vu,ku,beta: option remember: if K<20 then print(K, `shpild be at least 20 `): RETURN(FAIL): fi: beta:=17+12*sqrt(2): gu:=AnBn(a,b,c,d,e,K): if gu=FAIL then RETURN(FAIL): fi: lu:= [ [seq(gcd(numer(gu[1][i]),numer(gu[2][i])),i=1..nops(gu[1]))], [seq(lcm(denom(gu[1][i]),denom(gu[2][i])),i=1..nops(gu[1]))] ]: vu:=evalf([seq((log(lu[2][i+1])-log(lu[1][i+1]))/i,i=K-20..K)]): ku:=[ seq( (log(beta)-vu[i])/(log(beta)+vu[i]),i=1..nops(vu))]: ku:=evalf(ku,10): vu:=evalf(vu,10): [lu,vu,ku]: end: #TheoremZ3(a,b,c,d,e,K,P): Inputs rational numbers a,b,c,d,e, and a large positive integer K and P either 0 or a proposed #INTEGERATING factor, outputs #a theorem regarding the constant #Let [a1,a2,b1,b2,c1,c2,e]:=[b,c,e,a,a,c,d] #Int(Int(Int(x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)/(1-z+x*y*z)^(e+1)*(x*(1-x)*y*(1-y)*z*(1-z),x=0..1),y=0..1),z=0..1)) #divided by #Int(Int(Int(x^(a1)*(1-x)^(a2)*y^(b1)*(1-y)^(b2)*z^c1*(1-z)^c2/(1-z+x*y*z)^e,x=0..1),y=0..1),z=0..1)) #Either a suggested proof of irrationality or a way to compute it exponentially fast. #Try: #TheoremZ3(0,0,0,0,0,1000,0): TheoremZ3:=proc(a,b,c,d,e,K,P) local vu,n,N,gu,x,y,ope,lu,X,A,B,beta,F,C,E,A1,B1,delta,d1,ka,nuE,LCM,PP,i,WADIM: beta:=17+12*sqrt(2): gu:=CnDn(a,b,c,d,e,K): lu:=AnBn(a,b,c,d,e,K): ka:=GBCZ3ck(a,b,c,d,e)[1]: ope:=OPEZ3(a,b,c,d,e,n,N): vu:=ope[2]: ope:=ope[1]: if min(op(gu[3]))<0 then print(``): print(`----------------------------------------------------`): print(``): print(`Very fast Computation of rational approximations to the constant `): print( Int(Int(Int(x^(vu[1])*(1-x)^(vu[2])*y^(vu[3])*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(vu[7]+1),x=0..1),y=0..1),z=0..1) ): print(`divided by `): print( Int(Int(Int(x^(vu[1])*(1-x)^(vu[2])*y^(vu[3])*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(vu[7]),x=0..1),y=0..1),z=0..1) ): print(``): print(`that equals, up to`, Digits, ` digits `, GBCZ3ck(a,b,c,d,e)[1]): print(``): print(`By Shalosh B. Ekhad `): print(``): print(`Theorem: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`): print(``): print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0): print(``): print(`Subject to the initial conditions`): print(``): print(A(0)=lu[1][1],A(1)=lu[1][2]): print(``): print(B(0)=lu[2][1],B(1)=lu[2][2]): print(``): print(`Then`, A(n)/B(n), `approximates `): print(``): print(`The constant of the title, let's call it C `): print(``): print(`[That is approximately up to`, Digits, ` decimals `, GBCZ3ck(a,b,c,d,e)[1], `]`): print(``): print(``): print(`with an error that is OMEGA of`, (1/beta^2)^n): print(``): if identify(ka)<>ka then print(`Comment: Note that this constant appears to be `, identify(ka) ): print(`Prove it!`): fi: print(`Comment: while this sequence does not lead to an irrationality proof, for the record, the delta, happens to be roughly`): print(``): print(min(op(gu[3]))): print(``): print(`Proof, consider the Beukers type-integral F(n)`): print(``): print( Int( Int( Int( x^vu[1]*(1-x)^vu[2]*y^vu[3]*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(1+vu[7])*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n, x=0..1), y=0..1), z=0..1)): print(`divided by`): print( Int( Int( Int( x^vu[1]*(1-x)^vu[2]*y^vu[3]*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^vu[7], x=0..1), y=0..1), z=0..1) ): print(``): print(`Then `, F(0)=B(0)*C-A(0), F(1)=B(1)*C-A(1) ): print(``): print(`and F(n) also satisfies the above recurrence, thanks to Chrisoph Koutschan's amazing Matheamtics package HolonomicFunctions `): print(``): print(`Hence`, F(n)=B(n)*C-A(n)): print(``): print(`By a simple bound of the integrand, F(n) is OMEGA of`, 1/beta^n, ` and by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n): print(``): print(`Dividing by B(n) gives that A(n)/B(n)-C is OMEGA of `, 1/beta^(2*n) , `QED. `): else print(``): print(`----------------------------------------------------`): print(``): print(`Sketch of an Irrationality Proof of the constant `): print(``): print( Int(Int(Int(x^(vu[1])*(1-x)^(vu[2])*y^(vu[3])*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(vu[7]+1),x=0..1),y=0..1),z=0..1) ): print(`divided by `): print( Int(Int(Int(x^(vu[1])*(1-x)^(vu[2])*y^(vu[3])*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(vu[7]),x=0..1),y=0..1),z=0..1) ): print(``): print(`that equals, up to`, Digits, ` digits `, GBCZ3ck(a,b,c,d,e)[1]): print(``): print(`By Shalosh B. Ekhad `): print(``): print(`Theorem: The constant of the title , let's all it C`): print(`is irrational, with an irrationality measure`, 1+ (log(beta)+nu)/(log(beta)-nu), `for a certain number nu `): if P=0 then print(`that is approximately `, max(op(gu[2])) , ` yielding an irrationality measure that is approximately `, evalf(1+1/min(op(gu[3])),10)): print(``): print(`We hope that the reader can find nu exactly. `): else nuE:=AsyPpG(P): print(`that is EXACTLY`, nuE ): print(``): if not type(nuE,integer) then print( `that in decimals is `, evalf(nuE,10)): fi: print(``): print(` yielding an irrationality measure `): print( 1+ (log(beta)+nuE)/(log(beta)-nuE)): print(``): print(`that to 10 decimals, equals`): print(evalf( 1+ (log(beta)+nuE)/(log(beta)-nuE),10)): fi: if identify(ka)<>ka then print(`Comment: Note that this constant appears to be `, identify(ka) ): print(`Prove it!`): fi: print(`We need two lemmas `): print(``): print(`Lemma ONE: , let A(n), B(n), be two sequences of rational numbers that satisfy the second-order recurrence`): print(``): print( add(coeff(ope,N,i)*X(n+i),i=0..degree(ope,N)) =0): print(``): print(`Subject to the initial conditions`): print(``): print(A(0)=lu[1][1],A(1)=lu[1][2]): print(``): print(B(0)=lu[2][1],B(1)=lu[2][2]): print(``): print(`Then`, A(n)/B(n), `approximates the constant of the title, let's call it C`): print(``): print(`with an error that is OMEGA of`, (1/beta^2)^n): print(``): print(`Proof, consider the Beukers-type integral `): print(``): print( Int( Int( Int( x^vu[1]*(1-x)^vu[2]*y^vu[3]*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^(1+vu[7])*(x*(1-x)*y*(1-y)*z*(1-z)/(1-z+x*y*z))^n, x=0..1), y=0..1), z=0..1)): print(`divided by`): print( Int( Int( Int( x^vu[1]*(1-x)^vu[2]*y^vu[3]*(1-y)^vu[4]*z^vu[5]*(1-z)^vu[6]/(1-z+x*y*z)^vu[7], x=0..1), y=0..1), z=0..1) ): print(``): print(`Then `, F(0)=B(0)*C-A(0), F(1)=B(1)*C-A(1) ): print(``): print(`and F(n) also satisfies the above recurrence, thanks to the amazing Mathematica package HolonomicFunctions of Christoph Koutschan`): print(``): print(`Hence`, F(n)=B(n)*C-A(n)): print(``): print(`By a simple bound of the integrand, F(n) is OMEGA of`, 1/beta^n, ` and by the Poincare lemma, B(n) (and for that matter, A(n)) are OMEGA of`, beta^n): print(``): print(`Dividing by B(n) gives that A(n)/B(n)-C is OMEGA of `, 1/beta^(2*n) , `QED. `): print(``): print(`we now claim that the sequence of RATIONAL numbers A(n),B(n), can be multiplied by another sequence of rational numbers `): print(`E(n) such that both A(n)E(n) and B(n)E(n) are integers `): print(``): if P=0 then print(`Lemma TWO: There exists a sequence of rational numbers, whose prime factorizations consists of small primes, that hopefully`): print(`can be described (and proved) explicity, that we leave to the expert reader such that `): print(` A1(n):=E(n)A(n), B1(n):=E(n)B(n) are BOTH integers`): print(``): print(`Furthermore there exists a contant, nu, that hopefully the learned reader can determine such that E(n) is OMEGA of `, exp(nu*n) ): print(``): print(`The empircal values of nu for E(n) from`, K+1-nops(gu[2]), `to `, K+1, `are `): print(``): print(gu[2]): print(``): print(`Multiplying F(n) by E(n) we get `): print(``): print( E(n)*F(n)=B1(n)*C-A1(n)): print(``): print(`and this implies that `): print(``): print(abs(C-A1(n)/B1(n)) <= CONSTANT/B1(n)^(1+delta)): print(``): print(` where `, delta= (log(beta)-nu)/(log(beta) + nu) ): print(``): print(`Using the above values of nu for E(n) from`, K+1-nops(gu[3]), `to `, K+1, `the estimated deltas are `): print(``): print(gu[3]): print(``): print(`As you can see, they are all positive `): print(``): print(`We leave it to the reader to fill-in the details.`): print(``): else print(`Let E(n) be`): print(``): print(PrintPpG(P,n,LCM,PP)): print(``): print(`where LCM(1..m) is the least-common-multiple of the first m integers, and for 0FAIL then if min(op(mu[3]))>0 and max(op(mu[3]))<1 then gu:=[op(gu), gu1]: ALD:=ALD union {mu1}: fi: fi: fi: fi: od: od: od: od: od: gu: end: Hopefuls2:=proc(): [ [[0,0,0,0,0]], [[-1/2, -1/2, 1/2, 0, 0], [-1/2, 0, 0, -1/2, 0], [-1/2, 0, 0, 0, 1/2], [-1/2, 0 , 0, 1/2, 0], [-1/2, 0, 1/2, 0, 1/2], [0, -1/2, 0, -1/2, 0], [0, -1/2, 1/2, -1/ 2, 0], [0, -1/2, 1/2, 0, -1/2], [0, -1/2, 1/2, 0, 1/2], [0, -1/2, 1/2, 1/2, 0], [0, 0, 0, -1/2, -1/2], [0, 0, 0, -1/2, 1/2], [0, 0, 0, 1/2, 1/2], [0, 1/2, -1/2 , -1/2, 0], [0, 1/2, -1/2, 0, 1/2], [0, 1/2, 0, -1/2, 0], [0, 1/2, 0, 1/2, 0], [0, 1/2, 1/2, -1/2, 0], [0, 1/2, 1/2, 0, 1/2], [0, 1/2, 1/2, 1/2, 0], [1/2, -1/ 2, 1/2, 0, 0], [1/2, 0, 0, -1/2, 0], [1/2, 0, 0, 0, -1/2], [1/2, 0, 0, 0, 1/2], [1/2, 0, 0, 1/2, 0], [1/2, 0, 1/2, 0, -1/2], [1/2, 0, 1/2, 0, 1/2], [1/2, 1/2, -1/2, 0, 0], [1/2, 1/2, 1/2, 0, 0]], [[-2/3, -2/3, 1/3, -1/3, 0], [-2/3, -2/3, 1/3, 0, -1/3], [-2/3, -1/3, -1/3, 0, -1/3], [-2/3, -1/3, -1/3, 0, 2/3], [-2/3, -1/3, 0, -2/3, 0], [-2/3, -1/3, 0, -1 /3, 0], [-2/3, -1/3, 0, -1/3, 1/3], [-2/3, -1/3, 0, 1/3, 0], [-2/3, -1/3, 1/3, -1/3, 0], [-2/3, -1/3, 1/3, 0, 0], [-2/3, -1/3, 1/3, 0, 1/3], [-2/3, -1/3, 1/3, 2/3, 0], [-2/3, -1/3, 2/3, -1/3, 1/3], [-2/3, -1/3, 2/3, 0, -1/3], [-2/3, -1/3, 2/3, 0, 1/3], [-2/3, -1/3, 2/3, 0, 2/3], [-2/3, -1/3, 2/3, 2/3, 1/3], [-2/3, 0, -2/3, -1/3, 2/3], [-2/3, 0, -2/3, 0, 2/3], [-2/3, 0, -1/3, 0, 1/3], [-2/3, 0, 0 , -1/3, -2/3], [-2/3, 0, 0, -1/3, 0], [-2/3, 0, 0, -1/3, 1/3], [-2/3, 0, 0, 0, -2/3], [-2/3, 0, 0, 0, 1/3], [-2/3, 0, 0, 2/3, 0], [-2/3, 0, 0, 2/3, 1/3], [-2/ 3, 0, 1/3, -1/3, -2/3], [-2/3, 0, 1/3, -1/3, 1/3], [-2/3, 0, 1/3, -1/3, 2/3], [ -2/3, 0, 1/3, 0, 2/3], [-2/3, 0, 1/3, 2/3, 1/3], [-2/3, 0, 1/3, 2/3, 2/3], [-2/ 3, 0, 2/3, -2/3, 2/3], [-2/3, 0, 2/3, 0, -2/3], [-2/3, 0, 2/3, 0, 1/3], [-2/3, 0, 2/3, 0, 2/3], [-2/3, 0, 2/3, 1/3, 2/3], [-2/3, 1/3, 1/3, 0, -1/3], [-2/3, 1/ 3, 1/3, 0, 2/3], [-2/3, 2/3, -2/3, -1/3, 0], [-2/3, 2/3, -2/3, 0, 0], [-2/3, 2/ 3, -1/3, 0, -1/3], [-2/3, 2/3, -1/3, 0, 2/3], [-2/3, 2/3, 0, -2/3, 0], [-2/3, 2 /3, 0, -1/3, 0], [-2/3, 2/3, 0, 1/3, 0], [-2/3, 2/3, 0, 2/3, 0], [-2/3, 2/3, 1/ 3, -1/3, 0], [-2/3, 2/3, 1/3, 0, 0], [-2/3, 2/3, 1/3, 2/3, 0], [-2/3, 2/3, 2/3, 0, -1/3], [-2/3, 2/3, 2/3, 0, 2/3], [-1/3, -2/3, 0, -2/3, -1/3], [-1/3, -2/3, 0 , -2/3, 0], [-1/3, -2/3, 0, -2/3, 2/3], [-1/3, -2/3, 0, -1/3, 0], [-1/3, -2/3, 0, 2/3, 0], [-1/3, -2/3, 1/3, -2/3, -1/3], [-1/3, -2/3, 1/3, 0, -2/3], [-1/3, -\ 2/3, 1/3, 0, -1/3], [-1/3, -2/3, 1/3, 0, 1/3], [-1/3, -2/3, 1/3, 1/3, -1/3], [-\ 1/3, -2/3, 2/3, -2/3, 0], [-1/3, -2/3, 2/3, 0, -1/3], [-1/3, -2/3, 2/3, 0, 0], [-1/3, -2/3, 2/3, 0, 2/3], [-1/3, -2/3, 2/3, 1/3, 0], [-1/3, -1/3, -1/3, -2/3, 0], [-1/3, -1/3, 2/3, -2/3, 0], [-1/3, -1/3, 2/3, 0, -2/3], [-1/3, -1/3, 2/3, 0 , 1/3], [-1/3, -1/3, 2/3, 1/3, 0], [-1/3, 0, -2/3, 0, 2/3], [-1/3, 0, -1/3, -2/ 3, -1/3], [-1/3, 0, -1/3, -2/3, 1/3], [-1/3, 0, -1/3, 0, 1/3], [-1/3, 0, -1/3, 1/3, 1/3], [-1/3, 0, 0, -2/3, -1/3], [-1/3, 0, 0, -2/3, 0], [-1/3, 0, 0, -2/3, 2/3], [-1/3, 0, 0, 0, -1/3], [-1/3, 0, 0, 0, 2/3], [-1/3, 0, 0, 1/3, -1/3], [-1 /3, 0, 0, 1/3, 0], [-1/3, 0, 0, 1/3, 2/3], [-1/3, 0, 1/3, -1/3, 1/3], [-1/3, 0, 1/3, 0, -1/3], [-1/3, 0, 1/3, 0, 1/3], [-1/3, 0, 1/3, 0, 2/3], [-1/3, 0, 1/3, 2 /3, 1/3], [-1/3, 0, 2/3, -2/3, -1/3], [-1/3, 0, 2/3, -2/3, 1/3], [-1/3, 0, 2/3, -2/3, 2/3], [-1/3, 0, 2/3, 0, 1/3], [-1/3, 0, 2/3, 1/3, -1/3], [-1/3, 0, 2/3, 1 /3, 1/3], [-1/3, 0, 2/3, 1/3, 2/3], [-1/3, 1/3, -2/3, 0, -2/3], [-1/3, 1/3, -2/ 3, 0, 1/3], [-1/3, 1/3, -1/3, -2/3, 0], [-1/3, 1/3, -1/3, 0, 0], [-1/3, 1/3, -1 /3, 0, 2/3], [-1/3, 1/3, -1/3, 1/3, 0], [-1/3, 1/3, 0, -2/3, 0], [-1/3, 1/3, 0, -2/3, 2/3], [-1/3, 1/3, 0, -1/3, 0], [-1/3, 1/3, 0, 1/3, 0], [-1/3, 1/3, 0, 1/3 , 2/3], [-1/3, 1/3, 0, 2/3, 0], [-1/3, 1/3, 1/3, -2/3, 2/3], [-1/3, 1/3, 1/3, 0 , -2/3], [-1/3, 1/3, 1/3, 0, 1/3], [-1/3, 1/3, 1/3, 0, 2/3], [-1/3, 1/3, 1/3, 1 /3, 2/3], [-1/3, 1/3, 2/3, -2/3, 0], [-1/3, 1/3, 2/3, 0, 0], [-1/3, 1/3, 2/3, 0 , 2/3], [-1/3, 1/3, 2/3, 1/3, 0], [-1/3, 2/3, -1/3, 0, 1/3], [-1/3, 2/3, 2/3, 0 , -2/3], [-1/3, 2/3, 2/3, 0, 1/3], [0, -2/3, -2/3, -2/3, 0], [0, -2/3, 0, -2/3, -1/3], [0, -2/3, 0, -1/3, -1/3], [0, -2/3, 0, -1/3, 0], [0, -2/3, 0, 2/3, 0], [ 0, -2/3, 1/3, -2/3, 0], [0, -2/3, 1/3, -1/3, -2/3], [0, -2/3, 1/3, -1/3, 0], [0 , -2/3, 1/3, -1/3, 1/3], [0, -2/3, 1/3, 1/3, 0], [0, -2/3, 2/3, -2/3, -1/3], [0 , -2/3, 2/3, -2/3, 0], [0, -2/3, 2/3, -2/3, 2/3], [0, -2/3, 2/3, 0, -1/3], [0, -2/3, 2/3, 0, 2/3], [0, -2/3, 2/3, 1/3, -1/3], [0, -2/3, 2/3, 1/3, 0], [0, -2/3 , 2/3, 1/3, 2/3], [0, -1/3, -1/3, -2/3, -1/3], [0, -1/3, -1/3, -2/3, 0], [0, -1 /3, -1/3, -1/3, 0], [0, -1/3, 0, -2/3, -2/3], [0, -1/3, 0, -2/3, 0], [0, -1/3, 0, -2/3, 1/3], [0, -1/3, 0, -1/3, -2/3], [0, -1/3, 0, -1/3, 1/3], [0, -1/3, 0, 1/3, 0], [0, -1/3, 0, 1/3, 1/3], [0, -1/3, 1/3, -1/3, -2/3], [0, -1/3, 1/3, -1/ 3, 0], [0, -1/3, 1/3, -1/3, 1/3], [0, -1/3, 1/3, 0, -2/3], [0, -1/3, 1/3, 0, 1/ 3], [0, -1/3, 1/3, 2/3, 0], [0, -1/3, 1/3, 2/3, 1/3], [0, -1/3, 2/3, -2/3, -1/3 ], [0, -1/3, 2/3, -2/3, 0], [0, -1/3, 2/3, -2/3, 2/3], [0, -1/3, 2/3, -1/3, 0], [0, -1/3, 2/3, 1/3, -1/3], [0, -1/3, 2/3, 1/3, 0], [0, -1/3, 2/3, 1/3, 2/3], [0 , -1/3, 2/3, 2/3, 0], [0, 0, 0, -2/3, -1/3], [0, 0, 0, -2/3, 2/3], [0, 0, 0, -1 /3, -2/3], [0, 0, 0, -1/3, 1/3], [0, 0, 0, 1/3, -1/3], [0, 0, 0, 1/3, 2/3], [0, 0, 0, 2/3, -2/3], [0, 0, 0, 2/3, 1/3], [0, 1/3, -2/3, -2/3, 0], [0, 1/3, -2/3, -1/3, 0], [0, 1/3, -2/3, -1/3, 1/3], [0, 1/3, -1/3, -2/3, -1/3], [0, 1/3, -1/3, -2/3, 0], [0, 1/3, -1/3, -2/3, 2/3], [0, 1/3, -1/3, 0, -1/3], [0, 1/3, -1/3, 0, 2/3], [0, 1/3, -1/3, 1/3, 0], [0, 1/3, -1/3, 1/3, 2/3], [0, 1/3, 0, -2/3, -1/3] , [0, 1/3, 0, -2/3, 2/3], [0, 1/3, 0, -1/3, -1/3], [0, 1/3, 0, -1/3, 0], [0, 1/ 3, 0, -1/3, 2/3], [0, 1/3, 0, 1/3, -1/3], [0, 1/3, 0, 1/3, 2/3], [0, 1/3, 0, 2/ 3, 0], [0, 1/3, 0, 2/3, 2/3], [0, 1/3, 1/3, -2/3, 0], [0, 1/3, 1/3, -1/3, 0], [ 0, 1/3, 1/3, -1/3, 1/3], [0, 1/3, 1/3, 1/3, 0], [0, 1/3, 1/3, 2/3, 0], [0, 1/3, 1/3, 2/3, 1/3], [0, 1/3, 2/3, -2/3, -1/3], [0, 1/3, 2/3, -2/3, 0], [0, 1/3, 2/3 , -2/3, 2/3], [0, 1/3, 2/3, 0, -1/3], [0, 1/3, 2/3, 0, 2/3], [0, 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-4/5, 0], [-1/5, -3/5, 0, -3/5, -4/5], [-1/5, -3/5, 0, -3/5, 1/5], [-1/5, -3/5, 0, 1/5, 0], [-1/5, -3/5, 1/5, -2/5, 0], [-1/5, -2/5, 0, -3/5, 0], [-1/5, -2/5, 0, -2/5, -4/5], [-1/5, -2/5, 0, -2/5, 1/5], [-1/5, -2/5, 0, 2/5, 0], [-1/ 5, 1/5, -4/5, -3/5, 0], [-1/5, 1/5, 1/5, -3/5, 0], [-1/5, 1/5, 1/5, 2/5, 0], [-\ 1/5, 2/5, -4/5, -2/5, 0], [-1/5, 2/5, 0, -4/5, 0], [-1/5, 2/5, 0, -3/5, -4/5], [-1/5, 2/5, 0, -3/5, 1/5], [-1/5, 2/5, 0, 1/5, 0], [-1/5, 2/5, 0, 2/5, -4/5], [ -1/5, 2/5, 0, 2/5, 1/5], [-1/5, 2/5, 1/5, -2/5, 0], [-1/5, 2/5, 1/5, 3/5, 0], [ -1/5, 3/5, 0, -3/5, 0], [-1/5, 3/5, 0, -2/5, -4/5], [-1/5, 3/5, 0, -2/5, 1/5], [-1/5, 3/5, 0, 2/5, 0], [-1/5, 3/5, 0, 3/5, -4/5], [-1/5, 3/5, 0, 3/5, 1/5], [0 , -4/5, -4/5, -2/5, 0], [0, -4/5, -4/5, 3/5, 0], [0, -4/5, -2/5, -3/5, 0], [0, -4/5, -2/5, 2/5, 0], [0, -4/5, 0, -2/5, -3/5], [0, -4/5, 0, -1/5, -2/5], [0, -4 /5, 0, -1/5, 3/5], [0, -4/5, 0, 3/5, -3/5], [0, -4/5, 0, 4/5, -2/5], [0, -4/5, 0, 4/5, 3/5], [0, -4/5, 1/5, -3/5, -1/5], [0, -4/5, 1/5, -2/5, 0], [0, -4/5, 1/ 5, 3/5, 0], [0, -4/5, 3/5, -3/5, 0], [0, -4/5, 3/5, -1/5, -3/5], [0, -4/5, 3/5, -1/5, 2/5], [0, -4/5, 3/5, 2/5, 0], [0, -3/5, -4/5, -1/5, 0], [0, -3/5, -4/5, 4 /5, 0], [0, -3/5, -3/5, -4/5, 0], [0, -3/5, -3/5, 1/5, 0], [0, -3/5, 0, -4/5, -\ 1/5], [0, -3/5, 0, -2/5, -4/5], [0, -3/5, 0, -2/5, 1/5], [0, -3/5, 0, 1/5, -1/5 ], [0, -3/5, 0, 3/5, -4/5], [0, -3/5, 0, 3/5, 1/5], [0, -3/5, 1/5, -2/5, -1/5], [0, -3/5, 1/5, -2/5, 4/5], [0, -3/5, 1/5, -1/5, 0], [0, -3/5, 1/5, 4/5, 0], [0, -3/5, 2/5, -4/5, 0], [0, -3/5, 2/5, -1/5, -2/5], [0, -3/5, 2/5, -1/5, 3/5], [0, -3/5, 2/5, 1/5, 0], [0, -2/5, -2/5, -4/5, -3/5], [0, -2/5, -2/5, -1/5, 0], [0, -2/5, -2/5, 4/5, 0], [0, -2/5, -1/5, -4/5, 0], [0, -2/5, -1/5, -3/5, -4/5], [0, -2/5, -1/5, -3/5, 1/5], [0, -2/5, -1/5, 1/5, 0], [0, -2/5, 0, -3/5, -1/5], [0, -2/5, 0, -3/5, 4/5], [0, -2/5, 0, -1/5, -4/5], [0, -2/5, 0, -1/5, 1/5], [0, -2/ 5, 0, 2/5, -1/5], [0, -2/5, 0, 2/5, 4/5], [0, -2/5, 0, 4/5, -4/5], [0, -2/5, 0, 4/5, 1/5], [0, -2/5, 3/5, -4/5, -3/5], [0, -2/5, 3/5, -4/5, 2/5], [0, -2/5, 3/5 , -1/5, 0], [0, -2/5, 3/5, 1/5, -3/5], [0, -2/5, 3/5, 1/5, 2/5], [0, -2/5, 3/5, 4/5, 0], [0, -2/5, 4/5, -4/5, 0], [0, -2/5, 4/5, -3/5, -4/5], [0, -2/5, 4/5, -3 /5, 1/5], [0, -2/5, 4/5, 1/5, 0], [0, -2/5, 4/5, 2/5, -4/5], [0, -2/5, 4/5, 2/5 , 1/5], [0, -1/5, -3/5, -4/5, -2/5], [0, -1/5, -3/5, -4/5, 3/5], [0, -1/5, -3/5 , -2/5, 0], [0, -1/5, -3/5, 3/5, 0], [0, -1/5, -1/5, -3/5, 0], [0, -1/5, -1/5, -2/5, -4/5], [0, -1/5, -1/5, -2/5, 1/5], [0, -1/5, -1/5, 2/5, 0], [0, -1/5, 0, -4/5, -3/5], [0, -1/5, 0, -4/5, 2/5], [0, -1/5, 0, -3/5, -2/5], [0, -1/5, 0, -3 /5, 3/5], [0, -1/5, 0, 1/5, -3/5], [0, -1/5, 0, 1/5, 2/5], [0, -1/5, 0, 2/5, -2 /5], [0, -1/5, 0, 2/5, 3/5], [0, -1/5, 2/5, -4/5, -2/5], [0, -1/5, 2/5, -4/5, 3 /5], [0, -1/5, 2/5, -2/5, 0], [0, -1/5, 2/5, 1/5, -2/5], [0, -1/5, 2/5, 1/5, 3/ 5], [0, -1/5, 2/5, 3/5, 0], [0, -1/5, 4/5, -3/5, 0], [0, -1/5, 4/5, -2/5, -4/5] , [0, -1/5, 4/5, -2/5, 1/5], [0, -1/5, 4/5, 2/5, 0], [0, -1/5, 4/5, 3/5, -4/5], [0, -1/5, 4/5, 3/5, 1/5], [0, 1/5, -4/5, -3/5, -1/5], [0, 1/5, -4/5, -2/5, 0], [0, 1/5, -4/5, 3/5, 0], [0, 1/5, -2/5, -3/5, 0], [0, 1/5, -2/5, -1/5, -3/5], [0 , 1/5, -2/5, -1/5, 2/5], [0, 1/5, -2/5, 2/5, 0], [0, 1/5, 0, -2/5, -3/5], [0, 1 /5, 0, -2/5, 2/5], [0, 1/5, 0, -1/5, -2/5], [0, 1/5, 0, -1/5, 3/5], [0, 1/5, 0, 3/5, -3/5], [0, 1/5, 0, 3/5, 2/5], [0, 1/5, 0, 4/5, -2/5], [0, 1/5, 0, 4/5, 3/5 ], [0, 1/5, 1/5, -3/5, -1/5], [0, 1/5, 1/5, -3/5, 4/5], [0, 1/5, 1/5, -2/5, 0], [0, 1/5, 1/5, 2/5, -1/5], [0, 1/5, 1/5, 2/5, 4/5], [0, 1/5, 1/5, 3/5, 0], [0, 1 /5, 3/5, -3/5, 0], [0, 1/5, 3/5, -1/5, -3/5], [0, 1/5, 3/5, -1/5, 2/5], [0, 1/5 , 3/5, 2/5, 0], [0, 1/5, 3/5, 4/5, -3/5], [0, 1/5, 3/5, 4/5, 2/5], [0, 2/5, -4/ 5, -2/5, -1/5], [0, 2/5, -4/5, -2/5, 4/5], [0, 2/5, -4/5, -1/5, 0], [0, 2/5, -4 /5, 4/5, 0], [0, 2/5, -3/5, -4/5, 0], [0, 2/5, -3/5, -1/5, -2/5], [0, 2/5, -3/5 , -1/5, 3/5], [0, 2/5, -3/5, 1/5, 0], [0, 2/5, 0, -4/5, -1/5], [0, 2/5, 0, -4/5 , 4/5], [0, 2/5, 0, -2/5, -4/5], [0, 2/5, 0, -2/5, 1/5], [0, 2/5, 0, 1/5, -1/5] , [0, 2/5, 0, 1/5, 4/5], [0, 2/5, 0, 3/5, -4/5], [0, 2/5, 0, 3/5, 1/5], [0, 2/5 , 1/5, -2/5, -1/5], [0, 2/5, 1/5, -2/5, 4/5], [0, 2/5, 1/5, -1/5, 0], [0, 2/5, 1/5, 3/5, -1/5], [0, 2/5, 1/5, 3/5, 4/5], [0, 2/5, 1/5, 4/5, 0], [0, 2/5, 2/5, -4/5, 0], [0, 2/5, 2/5, -1/5, -2/5], [0, 2/5, 2/5, -1/5, 3/5], [0, 2/5, 2/5, 1/ 5, 0], [0, 2/5, 2/5, 4/5, -2/5], [0, 2/5, 2/5, 4/5, 3/5], [0, 3/5, -2/5, -4/5, -3/5], [0, 3/5, -2/5, -4/5, 2/5], [0, 3/5, -2/5, -1/5, 0], [0, 3/5, -2/5, 1/5, -3/5], [0, 3/5, -2/5, 1/5, 2/5], [0, 3/5, -2/5, 4/5, 0], [0, 3/5, -1/5, -4/5, 0 ], [0, 3/5, -1/5, -3/5, -4/5], [0, 3/5, -1/5, -3/5, 1/5], [0, 3/5, -1/5, 1/5, 0 ], [0, 3/5, -1/5, 2/5, -4/5], [0, 3/5, -1/5, 2/5, 1/5], [0, 3/5, 0, -3/5, -1/5] , [0, 3/5, 0, -3/5, 4/5], [0, 3/5, 0, -1/5, -4/5], [0, 3/5, 0, -1/5, 1/5], [0, 3/5, 0, 2/5, -1/5], [0, 3/5, 0, 2/5, 4/5], [0, 3/5, 0, 4/5, -4/5], [0, 3/5, 0, 4/5, 1/5], [0, 3/5, 3/5, -4/5, -3/5], [0, 3/5, 3/5, -4/5, 2/5], [0, 3/5, 3/5, -\ 1/5, 0], [0, 3/5, 3/5, 1/5, -3/5], [0, 3/5, 3/5, 1/5, 2/5], [0, 3/5, 3/5, 4/5, 0], [0, 3/5, 4/5, -4/5, 0], [0, 3/5, 4/5, -3/5, -4/5], [0, 3/5, 4/5, -3/5, 1/5] , [0, 3/5, 4/5, 1/5, 0], [0, 3/5, 4/5, 2/5, -4/5], [0, 3/5, 4/5, 2/5, 1/5], [0, 4/5, -3/5, -4/5, -2/5], [0, 4/5, -3/5, -4/5, 3/5], [0, 4/5, -3/5, -2/5, 0], [0, 4/5, -3/5, 1/5, -2/5], [0, 4/5, -3/5, 1/5, 3/5], [0, 4/5, -3/5, 3/5, 0], [0, 4/ 5, -1/5, -3/5, 0], [0, 4/5, -1/5, -2/5, -4/5], [0, 4/5, -1/5, -2/5, 1/5], [0, 4 /5, -1/5, 2/5, 0], [0, 4/5, -1/5, 3/5, -4/5], [0, 4/5, -1/5, 3/5, 1/5], [0, 4/5 , 0, -4/5, -3/5], [0, 4/5, 0, -4/5, 2/5], [0, 4/5, 0, -3/5, -2/5], [0, 4/5, 0, -3/5, 3/5], [0, 4/5, 0, 1/5, -3/5], [0, 4/5, 0, 1/5, 2/5], [0, 4/5, 0, 2/5, -2/ 5], [0, 4/5, 0, 2/5, 3/5], [0, 4/5, 2/5, -4/5, -2/5], [0, 4/5, 2/5, -4/5, 3/5], [0, 4/5, 2/5, -2/5, 0], [0, 4/5, 2/5, 1/5, -2/5], [0, 4/5, 2/5, 1/5, 3/5], [0, 4/5, 2/5, 3/5, 0], [0, 4/5, 4/5, -3/5, 0], [0, 4/5, 4/5, -2/5, -4/5], [0, 4/5, 4/5, -2/5, 1/5], [0, 4/5, 4/5, 2/5, 0], [0, 4/5, 4/5, 3/5, -4/5], [0, 4/5, 4/5, 3/5, 1/5], [1/5, -3/5, 0, -3/5, -1/5], [1/5, -3/5, 0, -3/5, 4/5], [1/5, -3/5, 0 , -2/5, 0], [1/5, -3/5, 0, 3/5, 0], [1/5, -2/5, -1/5, -3/5, 0], [1/5, -2/5, 0, -2/5, -1/5], [1/5, -2/5, 0, -2/5, 4/5], [1/5, -2/5, 0, -1/5, 0], [1/5, -2/5, 0, 4/5, 0], [1/5, -2/5, 4/5, -3/5, 0], [1/5, -2/5, 4/5, 2/5, 0], [1/5, -1/5, -1/5, -2/5, 0], [1/5, -1/5, 4/5, -2/5, 0], [1/5, -1/5, 4/5, 3/5, 0], [1/5, 2/5, 0, -3 /5, -1/5], [1/5, 2/5, 0, -3/5, 4/5], [1/5, 2/5, 0, -2/5, 0], [1/5, 2/5, 0, 2/5, -1/5], [1/5, 2/5, 0, 2/5, 4/5], [1/5, 2/5, 0, 3/5, 0], [1/5, 3/5, -1/5, -3/5, 0 ], [1/5, 3/5, -1/5, 2/5, 0], [1/5, 3/5, 0, -2/5, -1/5], [1/5, 3/5, 0, -2/5, 4/5 ], [1/5, 3/5, 0, -1/5, 0], [1/5, 3/5, 0, 3/5, -1/5], [1/5, 3/5, 0, 3/5, 4/5], [ 1/5, 3/5, 0, 4/5, 0], [1/5, 3/5, 4/5, -3/5, 0], [1/5, 3/5, 4/5, 2/5, 0], [1/5, 4/5, -1/5, -2/5, 0], [1/5, 4/5, -1/5, 3/5, 0], [1/5, 4/5, 4/5, -2/5, 0], [1/5, 4/5, 4/5, 3/5, 0], [2/5, -4/5, 0, -4/5, -2/5], [2/5, -4/5, 0, -4/5, 3/5], [2/5, -4/5, 3/5, -1/5, 0], [2/5, -2/5, 3/5, -4/5, 0], [2/5, -2/5, 3/5, 1/5, 0], [2/5, -1/5, 0, -4/5, 0], [2/5, -1/5, 0, -1/5, -2/5], [2/5, -1/5, 0, -1/5, 3/5], [2/5, -1/5, 0, 1/5, 0], [2/5, 1/5, -2/5, -1/5, 0], [2/5, 1/5, 0, -4/5, -2/5], [2/5, 1 /5, 0, -4/5, 3/5], [2/5, 1/5, 0, -2/5, 0], [2/5, 1/5, 0, 1/5, -2/5], [2/5, 1/5, 0, 1/5, 3/5], [2/5, 1/5, 0, 3/5, 0], [2/5, 1/5, 3/5, -1/5, 0], [2/5, 1/5, 3/5, 4/5, 0], [2/5, 3/5, -2/5, -4/5, 0], [2/5, 3/5, -2/5, 1/5, 0], [2/5, 3/5, 3/5, -\ 4/5, 0], [2/5, 3/5, 3/5, 1/5, 0], [2/5, 4/5, 0, -4/5, 0], [2/5, 4/5, 0, -1/5, -\ 2/5], [2/5, 4/5, 0, -1/5, 3/5], [2/5, 4/5, 0, 1/5, 0], [2/5, 4/5, 0, 4/5, -2/5] , [2/5, 4/5, 0, 4/5, 3/5], [3/5, -4/5, 0, -4/5, -3/5], [3/5, -4/5, 0, -4/5, 2/5 ], [3/5, -4/5, 0, -1/5, 0], [3/5, -4/5, 0, 4/5, 0], [3/5, -3/5, 2/5, -1/5, 0], [3/5, -1/5, -3/5, -4/5, 0], [3/5, -1/5, 0, -3/5, 0], [3/5, -1/5, 0, -1/5, -3/5] , [3/5, -1/5, 0, -1/5, 2/5], [3/5, -1/5, 0, 2/5, 0], [3/5, -1/5, 2/5, -4/5, 0], [3/5, -1/5, 2/5, 1/5, 0], [3/5, 1/5, 0, -4/5, -3/5], [3/5, 1/5, 0, -4/5, 2/5], [3/5, 1/5, 0, -1/5, 0], [3/5, 1/5, 0, 1/5, -3/5], [3/5, 1/5, 0, 1/5, 2/5], [3/5 , 1/5, 0, 4/5, 0], [3/5, 2/5, -3/5, -1/5, 0], [3/5, 2/5, 2/5, -1/5, 0], [3/5, 2 /5, 2/5, 4/5, 0], [3/5, 4/5, -3/5, -4/5, 0], [3/5, 4/5, -3/5, 1/5, 0], [3/5, 4/ 5, 0, -3/5, 0], [3/5, 4/5, 0, -1/5, -3/5], [3/5, 4/5, 0, -1/5, 2/5], [3/5, 4/5, 0, 2/5, 0], [3/5, 4/5, 0, 4/5, -3/5], [3/5, 4/5, 0, 4/5, 2/5], [3/5, 4/5, 2/5, -4/5, 0], [3/5, 4/5, 2/5, 1/5, 0], [4/5, -3/5, 0, -3/5, -4/5], [4/5, -3/5, 0, -\ 3/5, 1/5], [4/5, -3/5, 1/5, -2/5, 0], [4/5, -2/5, 0, -3/5, 0], [4/5, -2/5, 0, -\ 2/5, -4/5], [4/5, -2/5, 0, -2/5, 1/5], [4/5, -2/5, 0, 2/5, 0], [4/5, 1/5, 1/5, -3/5, 0], [4/5, 1/5, 1/5, 2/5, 0], [4/5, 2/5, -4/5, -2/5, 0], [4/5, 2/5, 0, -4/ 5, 0], [4/5, 2/5, 0, -3/5, -4/5], [4/5, 2/5, 0, -3/5, 1/5], [4/5, 2/5, 0, 1/5, 0], [4/5, 2/5, 0, 2/5, -4/5], [4/5, 2/5, 0, 2/5, 1/5], [4/5, 2/5, 1/5, -2/5, 0] , [4/5, 2/5, 1/5, 3/5, 0], [4/5, 3/5, 0, -3/5, 0], [4/5, 3/5, 0, -2/5, -4/5], [ 4/5, 3/5, 0, -2/5, 1/5], [4/5, 3/5, 0, 2/5, 0], [4/5, 3/5, 0, 3/5, -4/5], [4/5, 3/5, 0, 3/5, 1/5]] ]: end: #PrimesF(N,n,k): inputs a HUGE integer N that is a product of small primes, and a much smaller integer n #that generated N (via some process), outputs the list of length k, whose i-th entry is the list #of primes>=sqrt(n) that show up with exponent i, followed by the list of primes that #did not show up at all. Try: #lu:=op(Hopefuls2()[3][1]): N:=CnDn(op(lu),2000)[1][2][-1],2000): PrimesF(N,2000,6); PrimesF:=proc(N,n,k) local gu,PR,KULAM,T,i,gadol,katan,j,p: gu:=ifactors(N)[2]: if nops(gu)=1 then RETURN(gu): fi: PR:={}: for i from 1 to k do T[i]:=[]: od: for j from 1 to nops(gu) do if gu[j][1]>=evalf(sqrt(n)) then T[gu[j][2]]:=[op(T[gu[j][2]]),gu[j][1]]: PR:=PR union {gu[j][1]}: fi: od: gadol:=max(op(PR)): katan:=min(op(PR)): KULAM:={}: p:=nextprime(katan): KULAM:={p}: while p<=gadol do p:=nextprime(p): KULAM:=KULAM union {p}: od: PR:=KULAM minus PR: [ [seq(T[i],i=1..k)],PR]: end: #Lc(n): the lcm of 1, ..., n. Try: Lc(1000); Lc:=proc(n) local i: lcm(seq(i,i=1..n)): end: #Pp(a,b,n,m): Given rational numbers a and b between 0 and 1 and positive integers m and n #it is the product of all primes p less than m such that frac{n/p} is between a and b Pp:=proc(a,b,n,m) local p,gu: p:=2: gu:=1: p:=2: while p= a and frac(n/p)= a and frac(n/p)1 then RETURN(FAIL): fi: alpha:=alpha[1]: c:=1: N1:=1: for i from 1 to nops(gu) do if abs(gu[i][2]-alpha)<10 then c:=c*gu[i][1]: N1:=N1*gu[i][1]^gu[i][2]: fi: od: ka:=evalf(log(N1)/(n*log(c))): vu:=RoundRat(ka,K,eps): if vu<>FAIL then RETURN( [[c,vu], N/c^trunc(vu*n)] ): fi: FAIL: end: #RoundRat(ka,K,eps): inputs a number and a small postitive integer K, finds a rational number a/b with #denominator less than K such that abs(ka-a/b)0.01 do od: if i=11 then RETURN(FAIL): fi: for j from 1 to 10 while S1[nops(S1)-j+1]-S1[nops(S1)-j]>0.05 do od: if j=11 then RETURN(FAIL): fi: S1:=[op(i..nops(S1)-j,S1)]: a:=RoundRat(S1[1],K,eps/5): if a=FAIL then RETURN(FAIL): fi: b:=RoundRat(S1[-1],K,eps/5): if b=FAIL then FAIL: else [c,[a,b]]: fi: end: #ExtractPp(N,n,K1,K2,r,eps): inputs a large integer N belonging to some much smalller integer n (N is the n-th member of some sequence, in real life) #Conjectures an approrixmation to it in the form lcm(1..A*n)*Pp(a1,b1,n,n*c1)^1*Pp(a2,b2,n,n*c2)^2*... #The output is [A,[a1,b1,c1],[a2,b2,c2], ...]. K1,K2,r, eps are guessing parameters. For example, try: #ExtractPp(CnDn(0,1/4,0,3/4,1/2,2000)[1][2][-1],2000,10,5,0.05); ExtractPp:=proc(N,n,K1,K2,r,eps) local A,gu,N1,mu,i,vu,alpha,Vu: gu:=ifactors(N)[2]: A:=round(gu[-1][1]/n): if evalf(abs(gu[-1][1]/n-A))>eps then RETURN(FAIL): fi: for alpha from 1 while denom(N/Lc(A*n)^alpha)<10^(3 *K1) do od: alpha:=alpha-1: N1:=N/Lc(A*n)^alpha: mu:=[A,alpha]: if N1<10^(2*K1) then RETURN( [ [mu,[]] ,N1]): fi: gu:=PrimesF(numer(N1),n,r)[1]: Vu:=[]: for i from 1 to nops(gu) do if nops(gu[i])>50 then vu:=GuessPp1(gu[i],n,K2,eps): if vu=FAIL or vu[1]=FAIL then RETURN(FAIL): fi: Vu:=[op(Vu),vu]: N1:=N1/Pp(op(vu[2]),n,trunc(vu[1]*n))^i: fi: od: [[mu,Vu],N1]: end: Guess:=proc(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC,CUTOFF) local gu, i: gu:=Guess1(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC): if gu=FAIL then RETURN(FAIL): fi: if max(seq(numer(gu[3][i]),i=1..nops(gu[3])))>CUTOFF or max(seq(denom(gu[3][i]),i=1..nops(gu[3])))>CUTOFF then RETURN(FAIL): fi: gu: end: #Guess1(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC): Guesses an "Integrating Factor" for the irrationality of GBCZ3ck(a,b,c,d,e) (q.v.) #with guessing parameters K1,K2,r,eps. It inestigates 1000,2000, M*1000 #It returns the approximation to Dn followed by the approximation of Cn #It P is a symbol denoting Pp(a,b,n,m) (q.v.) and LC(a) denots lcm(1...a) #It also returns the deviations #Try: #Guess1(0,1/4,0,3/4,1/2,10,10,6,0.05,4,n,P,LC): Guess1:=proc(a,b,c,d,e,K1,K2,r,eps,M,n,P,LC) local mu,gu,lu,i,B,j,vu,C: for i from 1 to M do mu:=CnDn(a,b,c,d,e,1000*(i+1))[1]: gu[i]:=ExtractPower(mu[1][-1],1000*(i+1),K1,eps) : if gu[i]=FAIL then RETURN(FAIL): else gu[i]:=gu[i][1]: fi: lu[i]:=ExtractPp(mu[2][-1],1000*(i+1),K1,K2,r,eps) : if lu[i]=FAIL then RETURN(FAIL): else lu[i]:=lu[i][1]: fi: od: if nops({seq(gu[i],i=1..M)})<>1 or nops({seq(lu[i],i=1..M)})<>1 then RETURN(FAIL): fi: gu:=gu[1]: lu:=lu[1]: B:=1/gu[1]^(gu[2]*n): B:=B*LC(lu[1][1]*n)^lu[1][2]: vu:=lu[2]: for j from 1 to nops(vu) do B:=B*P(op(vu[j][2]),n,vu[j][1]*n)^j: od: C:=[seq(CnDn(a,b,c,d,e,i*1000)[1][2][-1]/CnDn(a,b,c,d,e,i*1000)[1][1][-1]/EvalPp([gu,lu], i*1000),i=1..3)]: [[gu,lu],B,C]: end: #EvalPp(P,n1): evaluate a Pp object at n=n1. Try: #EvalPp([[3, 3], [[3, 1], [[29/10, [1/3, 2/3]], [3/2, [2/3, 1]]]]]); EvalPp:=proc(P,n1) local gu,lu,i: if not (type(P,list) and nops(P)=2 and nops(P[1])=2 and nops(P[2])=2 and nops(P[2][1])=2) then print(`Bad input`): RETURN(FAIL): fi: gu:=1/P[1][1]^(P[1][2]*n1): gu:=gu*Lc(P[2][1][1]*n1)^P[2][1][2]: lu:=P[2][2]: for i from 1 to nops(lu) do gu:=gu*Pp(op(lu[i][2]),n1, trunc(lu[i][1]*n1))^i: od: gu: end: EvalPp:=proc(P,n1) local gu,lu,i: if not (type(P,list) and nops(P)=2 and nops(P[1])=2 and nops(P[2])=2 and nops(P[2][1])=2) then print(`Bad input`): RETURN(FAIL): fi: gu:=1/P[1][1]^(P[1][2]*n1): gu:=gu*Lc(P[2][1][1]*n1)^P[2][1][2]: lu:=P[2][2]: for i from 1 to nops(lu) do gu:=gu*Pp(op(lu[i][2]),n1, trunc(lu[i][1]*n1))^i: od: gu: end: #MyID(C,F,N): Given a constant C in floating point and a famous constant F (like log(2)) and a positive integer N #tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ #MyID(evalf((log(2)-2)/(2*log(2)+3)),log(2),100); MyID:=proc(C,F,N) local gu,mu,i: gu:=IntegerRelations[PSLQ]([1,evalf(C),evalf(F),evalf(C*F)]): if max(seq(abs(gu[i]),i=1..4)) >N then RETURN(FAIL): fi: mu:=-(gu[1]+gu[3]*F)/(gu[2]+gu[4]*F): if abs(evalf(mu-C))>1/10^(Digits-3) then RETURN(FAIL): fi: mu: end: #MyIDs(C,F,F0,N): Given a constant C in decimals and another constant #let's call it F, whose floating-point if F0 # and a positive integer N #tries to express C as (a*F+b)/(c*F+d) for a,b,c,d from -N to N using PSLQ #MyIDs(evalf((log(2)-2)/(2*log(2)+3)),log(2),evalf(log(2)),100); MyIDs:=proc(C,F,F0,N) local gu,mu,i: gu:=IntegerRelations[PSLQ]([1,evalf(C),evalf(F0),evalf(C*F0)]): if max(seq(abs(gu[i]),i=1..4)) >N then RETURN(FAIL): fi: mu:=-(gu[1]+gu[3]*F)/(gu[2]+gu[4]*F): if abs(evalf(subs(F=F0,mu)-C))>1/10^(Digits-7) then RETURN(FAIL): fi: mu: end: # PpLimit(a, b, r): Return the limit of log(Pp(a, b, n, rn)) / n as n tends to # infinity. # Try: # evalf([log(Pp(1/4, 3/4, 1000, 2 * 1000)) / 1000, PpLimit(1/4, 3/4, 2)]); # evalf([log(Pp(3/4, 1, 1000, 4 * 1000 / 3)) / 1000, PpLimit(3/4, 1, 4/3)]); # evalf([log(Pp(1/10, 1/2, 1000, 5 * 1000)) / 1000, PpLimit(1/10, 1/2, 5)]); # evalf([log(Pp(1/10, 2/10, 2000, 3 * 2000)) / 2000, PpLimit(1/10, 2/10, 3)]); PpLimit:=proc(a, b, r) local t: if r=0 then return 0: fi: if a = 0 then return Psi(b) + max(r, 1 / b) - limit(Psi(t) + 1 / t, t = 0): fi: Psi(b) - Psi(a) - ifelse(a < 1 / r, 1 / a - max(r, 1 / b), 0): end: #SortC(L,N,r): Given a list of pentuples, L, and a positive integer N, and a symbol r #divides them into classes such that GBCZ3ck(op(L[i]))[1] are related to #each other (conjecturally) by a fractional-linear transfomation with integers less than N in absolute value #The output is a list of lists such that the first entry is the main pentuple, followed by the symbol r[i], #its decimal approximation, followed by the tuples, and the expression in terms of r[i] #Try: #SortC(Hopefuls2()[2],10000,r); SortC:=proc(L,N,r) local i,S1,Rf,lu1,lu2,co,T,ka: S1:=convert(L,set): co:=0: while S1<>{} do co:=co+1: lu1:=S1[1]: Rf:=GBCZ3ck(op(lu1))[1]: T[co]:=[]: for lu2 in S1 do ka:=MyIDs(GBCZ3ck(op(lu2))[1],r[co],Rf,N): if ka<>FAIL then T[co]:=[op(T[co]),[lu2,ka]]: S1:=S1 minus {lu2}: fi: od: od: [seq(T[i],i=1..co)]: end: #AsyPp(P): inuts a Pp expression of the form outputted by Guess, and outputs the #EXACT value of limit of log( EvalPp(P,n))/n as n goes to infinity. Try #AsyPp([[3, 3], [[3, 1], [[3, [1/3, 2/3]], [3/2, [2/3, 1]]]]]); AsyPp:=proc(P) local gu,lu,i: if not (type(P,list) and nops(P)=2 and nops(P[1])=2 and nops(P[2])=2 and nops(P[2][1])=2) then print(`Bad input`): RETURN(FAIL): fi: #PpLimit := proc(a, b, r) #gu:=1/P[1][1]^(P[1][2]*n1): gu:=-P[1][2]*log(P[1][1]): gu:=gu+P[2][1][1]*P[2][1][2]: lu:=P[2][2]: for i from 1 to nops(lu) do gu:=gu+i*PpLimit(op(lu[i][2]),lu[i][1]): od: gu: end: #The limit of log PpG(a,b,r1*n,r2*n,C,M)/n as n goes to infinity. Try #PpGlimit(1/3,2/3,3/2,3,{2},3); PpGlimit:=proc(a,b,r1,r2,C,M) (PpLimit(a,b,r2)-PpLimit(a,b,r1))*nops(C)/numtheory[phi](M): end: #AsyPpG(P): inuts a Pp expression of the form outputted by Guess, and outputs the #EXACT value of limit of log( EvalPpG(P,n))/n as n goes to infinity. Try #AsyPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1], [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ] ); AsyPpG:=proc(P) local gu,lu,i: if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and (P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6} )) then print(`Bad input`): RETURN(FAIL): fi: gu:=-P[1][2]*log(P[1][1]): gu:=gu+P[2][1]*P[2][2]: lu:=P[3]: for i from 1 to nops(lu) do gu:=gu+PpGlimit(op(lu[i])): od: gu: end: #EvalPpG(P,n1): evaluate a PpG object at n=n1. #For GBCZ3ck(0,0,0,0,0) (alias Zeta(3)) #EvalPpG([[1, 1], [1, 3],[] ],1000 ); #For GBCZ3ck(0,0,0,1/3,2/3), try: #EvalPpG([[3, 3], [3, 1], [ [1/3,1,0,3/2,{0},1], [1/3,1,3/2,3,{2},3],[2/3,1,0,3/2,{0},1]] ],1000 ); #For GBCZ3ck(0,1/4,0,3/4,1/2), try: #EvalPpG([[2, 6], [4, 1], [ [1/4,3/4,0,2,{0},1], [3/4,1,0,4/3,{0},1], [3/4,1,0,4/3,{0},1] ] ],1000 ); EvalPpG:=proc(P,n1) local gu,lu,i,lu1: if not (type(P,list) and nops(P)=3 and nops(P[1])=2 and nops(P[2])=2 and (P[3]=[] or {seq(nops(P[3][i]),i=1..nops(P[3]))}={6}) ) then print(`Bad input`): RETURN(FAIL): fi: gu:=1/P[1][1]^(P[1][2]*n1): gu:=gu*Lc(P[2][1]*n1)^P[2][2]: lu:=P[3]: for i from 1 to nops(lu) do lu1:=lu[i]: gu:=gu*PpG(lu1[1],lu1[2],n1,trunc(lu1[3]*n1),trunc(lu1[4]*n1),lu1[5],lu1[6] ): od: gu: end: #Fs(S,n): Given a set S and a positive integer n finds sort([seq(evalf(frac(n/p),10),p in S)]); Fs:=proc(S,n) local p: sort([seq(evalf(frac(n/p),10),p in S)]); end: #Seg1(L,a,b): Given a list L and integers ab then RETURN(FAIL): fi: for i from 1 while L1[i]