#OK to post homework
#Wanying Rao, 03/28/2021, Assignment 17
#3
CheckMordell := proc(N) local L,p,q:

L := RseqFF(24,N^2):

for p from 2 to N do
 for q from p+1 to N do
  if type(p,prime) and type(q,prime) then
   if L[p]*L[q] <> L[p*q] then
    RETURN(false):
   fi:
  fi:
 od:
od:

RETURN(true):
end:

#4
CheckDeligne := proc(N) local i,L:

L := [seq(infinity,i=1..N)]:

for i from 2 to N do
 if type(i, prime) then
  L[i] := abs(RseqFF(24,i)[i])/(2*i^(11/2)):
 fi:
od:

RETURN([min[index](L),min(L)]):
end:

#From C17.txt:
#JC(P,x,m,K): inputs a polynomial P in the variable x, a pos. integer m
#and a pos. integer K, outputs the coeff. of x^n from n=0 to n=K
#In other words taylor(P^m,x=0,K+1):
#P^m=a(m,0)+a(m,1)*x+a(m,2)*x^2+...
#P=p[0]+p[1]*x+...+p[L]*x^L
#P=3+2*x+5*x^2
#p[0]=3, p[1]=2, p[2]=5
#a(m,k)=1/(k*p0)*Sum(p[i]*((m+1)*i-k)*a(m,k-i),i=1..L):
JC:=proc(P,x,m,K) local L,p0,M,ng,k:
L:=degree(P,x):
p0:=coeff(P,x,0):
M:=[p0^m]:

for k from 1 to K do
#ng is a(m,k):
ng:= add(coeff(P,x,i)*((m+1)*i-k)*M[k-i+1],i=1..min(L,k))/(k*p0): 
M:=[op(M),ng]:
od:
  
end:

#RseqFF(r,N): a yet Fast version of RseqF(r,N) (see above) using the J.C.P. Miller recurrence
RseqFF:=proc(r,N) local f,i,q:
f:=1:
for i from 1 while (3*i+1)*i/2<=N do
 f:=f+(-1)^i*q^((3*i-1)*i/2) +(-1)^i*q^((3*i+1)*i/2):
od:

 
JC(f,q,r,N-1):

end: