A sequence of Diophantine Approximations to Pi that implies its irrationality With a fully rigorous crude upper bound on the irrationality measure of, 10.747747459921685428 and a not-yet-rigorous smaller upper bound around, 7.1674669283251400761 By Shalosh B. Ekhad Consider the following Salikhov-type integral, let's call it E(n) 4 + 2 I / | (4 n) (4 n) (4 n) -I | (x - 4 + 2 I) (x - 4 - 2 I) (x - 5) | / 4 - 2 I (4 n) (4 n) / (6 n + 1) (6 n + 1) (x - 6 + 2 I) (x - 6 - 2 I) / (x (x - 10) ) dx / It is readily seen that E(n) can be written as E(n)=C(n) + D(n)*Pi For certain sequences of RATIONAL numbers C(n), D(n) It follows from the amazing Almkvist-Zeilberger algorithm that E(n), and hen\ ce C(n) and D(n) satisfy the following third-order linear recurrence equation with polynomial coefficients. -524288 (4 n + 11) (4 n + 7) (4 n + 3) (2 n + 3) (1 + 2 n) (4 n + 9) (4 n + 5) 10 9 (4 n + 1) (n + 2) (n + 1) (60867901643065920 n + 1402311025222353792 n 8 7 + 14470723426942041168 n + 88063190071996560720 n 6 5 + 349936747144283090428 n + 948558058323541387488 n 4 3 + 1775910450337084423423 n + 2267102173453550031081 n 2 + 1888167286839295642309 n + 926219087924595329151 n + 203162908348831933656) F(n) + 8192 (4 n + 9) (4 n + 5) (2 n + 3) 14 (4 n + 11) (4 n + 7) (n + 2) (281201427556341093434880 n 13 12 + 7322090504896867559228928 n + 87209831457597091015623552 n 11 10 + 629049661611019429946258496 n + 3066017551085139101315515040 n 9 8 + 10667054261779260766882889376 n + 27274872903904098452693111000 n 7 6 + 51973074104501208047402674196 n + 74012716893560451515633769104 n 5 4 + 78199285791627472057084570886 n + 60180273781064951742729517935 n 3 2 + 32610873164161232620698772253 n + 11722242036123894963902865396 n + 2492713092388358114498630880 n + 235757247529875283004663808) F(n + 1) 18 - 24 (4 n + 9) (4 n + 11) (1808026060746077792794776775680 n 17 + 59734643466614275427914449543168 n 16 + 923568741816213162733177441676544 n 15 + 8875780866018363938849235850717440 n 14 + 59390274225322200281022686012242560 n 13 + 293694785522776558009548044260616448 n 12 + 1111840405134721987490664006716858912 n 11 + 3292856121031828128212380672718441248 n 10 + 7729456635439513427032192850349082460 n 9 + 14477929153168505016088675653760022208 n 8 + 21673847317267166448876296503356616449 n 7 + 25847657558552536448884848497546441949 n 6 + 24349214613589153208449822405468841905 n 5 + 17852197826047775298808554253368388703 n 4 + 9949489121137459919248608014050611730 n 3 + 4061016510751343038717843113897589788 n 2 + 1141134289722614824335076469108319720 n + 196610384848069665712806514029351648 n + 15599824063034945611316134148893440) F(n + 2) + 9 (8 n + 23) (6 n + 17) (3 n + 8) (8 n + 21) (2 n + 5) (8 n + 19) (3 n + 7) (6 n + 13) (8 n + 17) 10 9 (n + 3) (60867901643065920 n + 793632008791694592 n 8 7 + 4588979773878823440 n + 15476451367297057488 n 6 5 + 33662805821050442764 n + 49256533024980280584 n 4 3 + 49001423416714364371 n + 32647130987599531629 n 2 + 13903215538367178616 n + 3407705309334959580 n + 364163199174534672) F(n + 3) = 0 and in Maple format -524288*(4*n+11)*(4*n+7)*(4*n+3)*(2*n+3)*(1+2*n)*(4*n+9)*(4*n+5)*(4*n+1)*(n+2)* (n+1)*(60867901643065920*n^10+1402311025222353792*n^9+14470723426942041168*n^8+ 88063190071996560720*n^7+349936747144283090428*n^6+948558058323541387488*n^5+ 1775910450337084423423*n^4+2267102173453550031081*n^3+1888167286839295642309*n^ 2+926219087924595329151*n+203162908348831933656)*F(n)+8192*(4*n+9)*(4*n+5)*(2*n +3)*(4*n+11)*(4*n+7)*(n+2)*(281201427556341093434880*n^14+ 7322090504896867559228928*n^13+87209831457597091015623552*n^12+ 629049661611019429946258496*n^11+3066017551085139101315515040*n^10+ 10667054261779260766882889376*n^9+27274872903904098452693111000*n^8+ 51973074104501208047402674196*n^7+74012716893560451515633769104*n^6+ 78199285791627472057084570886*n^5+60180273781064951742729517935*n^4+ 32610873164161232620698772253*n^3+11722242036123894963902865396*n^2+ 2492713092388358114498630880*n+235757247529875283004663808)*F(n+1)-24*(4*n+9)*( 4*n+11)*(1808026060746077792794776775680*n^18+59734643466614275427914449543168* n^17+923568741816213162733177441676544*n^16+8875780866018363938849235850717440* n^15+59390274225322200281022686012242560*n^14+ 293694785522776558009548044260616448*n^13+1111840405134721987490664006716858912 *n^12+3292856121031828128212380672718441248*n^11+ 7729456635439513427032192850349082460*n^10+ 14477929153168505016088675653760022208*n^9+ 21673847317267166448876296503356616449*n^8+ 25847657558552536448884848497546441949*n^7+ 24349214613589153208449822405468841905*n^6+ 17852197826047775298808554253368388703*n^5+ 9949489121137459919248608014050611730*n^4+4061016510751343038717843113897589788 *n^3+1141134289722614824335076469108319720*n^2+ 196610384848069665712806514029351648*n+15599824063034945611316134148893440)*F(n +2)+9*(8*n+23)*(6*n+17)*(3*n+8)*(8*n+21)*(2*n+5)*(8*n+19)*(3*n+7)*(6*n+13)*(8*n +17)*(n+3)*(60867901643065920*n^10+793632008791694592*n^9+4588979773878823440*n ^8+15476451367297057488*n^7+33662805821050442764*n^6+49256533024980280584*n^5+ 49001423416714364371*n^4+32647130987599531629*n^3+13903215538367178616*n^2+ 3407705309334959580*n+364163199174534672)*F(n+3) = 0 The initial conditions are as follows Pi 18662336 Pi 6156093056 E(0) = - ----, E(1) = - ----------- + ----------, 20 5 525 6331240948773888 Pi 11635755330873327616 E(2) = - ------------------- + -------------------- 5 2925 and in Maple notation E(0) = -1/20*Pi, E(1) = -18662336/5*Pi+6156093056/525, E(2) = -6331240948773888 /5*Pi+11635755330873327616/2925 Using this recurrence, we can compute many values, and use them to ESTIMATE \ the implied bound irrationality measure. Using the values from n=, 990, to M=, 1000, and taking the largest value yields the following estimate 7.0836841941481932631 We can use the Poincare lemma to find the asymptotic behavior of the exponti\ ally growing C(n), D(n) and the exponentially decaying E(n). The constant-coefficient approximation to the above recurrence has indicial \ polynomial 3 2 -11664 N + 5569518206457 N - 9461481472 N + 4194304 and in Maple format -11664*N^3+5569518206457*N^2-9461481472*N+4194304 9 whose largest root let's call it a, is, 0.47749641687561601769 10 , and absolute value of the two smaller roots, let's call it b is, 0.00086780298105928038750 It follows that, up to polynomial correction that will get out in the wash, \ C(n) and D(n) are OMEGA(a^n) and E(n) is O(b^n) We now need a divisibility lemma that is left to the reader Let d(n)=lcm(1..n) Lemma: n n C1(n) = d(8 n) (1/32) C(n), D1(n) = d(8 n) (1/32) D(n) are integers Defining n E1(n) = d(8 n) (1/32) E(n) We have E1(n) = C1(n) + Pi D1(n) where C1(n) and D1(n) are INTEGERS Since, famously, d(n)=OMEGA(e^n), we have C1(n) and D1(n) are OMEGA of n n exp(8 n) (1/32) a and E1(n) is O of n n exp(8 n) (1/32) b It follows that E1(n)=max(C1(n),D1(n))^(-delta) where delta equals ln(b) + 8 - 5 ln(2) - ------------------- ln(a) + 8 - 5 ln(2) 9 recall that a is , 0.47749641687561601769 10 , and b is , 0.00086780298105928038750 and delta is .10258780339884124572 implying the rigorous, but crude upper bound for the irrationality measure, \ 1+1/delta that equals 10.747747459921685428 confirming the first part of the statement in the title We now need a harder lemma, that we confess that we can't do, but you the re\ ader may be able to Harder Lemma: There exists a rigorously proved constant K1 such that gcd(C1(n),D1(n))=O(exp(K1*n)) Once we find such a constant K1 the delta can be improved to By looking at the smallest value of log(gcd(C1(n),D1(n))/n for n between n=, 500, and , 1000, we estimate that K1 can be taken to be 1.2564291372372349421 and the better delta is .16214112075856135400 that implies an irrationality measure, 1+1/delta that equals 7.1674669283251400761 ----------------------------------------- This ends this paper that took, 20.295, to generate.