A sequence of Diophantine Approximations to Pi that implies its irrationality With a fully rigorous crude upper bound on the irrationality measure of, 9.7148240138850430652 and a not-yet-rigorous smaller upper bound around, 7.4929033960782814609 By Shalosh B. Ekhad Consider the following Salikhov-type integral, let's call it E(n) 4 + 2 I / | (6 n) (6 n) (6 n) -I | (x - 4 + 2 I) (x - 4 - 2 I) (x - 5) | / 4 - 2 I (6 n) (6 n) / (10 n + 1) (10 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. -127401984 (6 n + 17) (6 n + 11) (6 n + 5) (3 n + 5) (3 n + 2) (2 n + 3) (2 n + 1) (3 n + 4) (3 n + 1) (6 n + 13) (6 n + 7) (6 n + 1) (n + 2) 12 11 (n + 1) (31631151990069989341341760 n + 823419236582991893828142640 n 10 9 + 9771603677661324385456449360 n + 69890086912088964626195669400 n 8 7 + 335495829594607960813857849996 n + 1138521859146775520973744447423 n 6 5 + 2800226928403038874957054123859 n + 5028595180854507257622282060236 n 4 3 + 6542531718883433282475330075342 n + 6013469985835877898194794695771 n 2 + 3705745147593721221432173071323 n + 1374471665328843740407971742290 n + 232006820127400470851515693800) F(n) - 552960 (6 n + 17) (6 n + 11) (3 n + 5) (2 n + 3) (3 n + 4) (6 n + 13) (6 n + 7) (n + 2) ( 18 9587470239694542479580549597121190400 n 17 + 287930023598216908999507426488571747200 n 16 + 4021877375756635798330051427520713704000 n 15 + 34689677494238497057571063144228984647200 n 14 + 206851587187304227315977540614033360214800 n 13 + 904622068102959212353476594649472661237400 n 12 + 3003719974100166273302912427584923554247500 n 11 + 7733432024373755308471465432599999473960150 n 10 + 15630008693262114542137087593371584435138001 n 9 + 24946965741709714320490326701337465872205285 n 8 + 31467826499092483581505369869871230925508242 n 7 + 31237677151890886255508800106752972249110290 n 6 + 24172411400699527442985964446510965175946752 n 5 + 14349771599634571971675180709913315803026742 n 4 + 6374433012509013669326294664696915832278939 n 3 + 2038407166254294152478752122920339689752413 n 2 + 440375133986396777135117827809979390801206 n + 57143812531234319584459751979012103157880 n + 3339405646600612898553477526783997337600) F(n + 1) - 3000 (10 n + 19) (6 n + 17) (10 n + 17) (10 n + 13) (6 n + 13) (10 n + 11) ( 20 62663602584859049144837469021671297399655680 n 19 + 2132561954240463724476820857712871596901568960 n 18 + 34082773774697012366585011375172505086093834560 n 17 + 339955581485702825273120443339315846765469445840 n 16 + 2372056594220852752944331289033277328223925784608 n 15 + 12299689181130742103419393973497841048072376747780 n 14 + 49142542460189050302672696923848522827303035982980 n 13 + 154804044204977372678319835669599317114483980302507 n 12 + 390149378455261315866151510206244436577194331589795 n 11 + 793690570437899682848898506496595829218322834548764 n 10 + 1309021009777480827729405852420047916482250598462710 n 9 + 1751269657988698314556992277015122217112241464501391 n 8 + 1894589397459230912118646639760883070265296031507891 n 7 + 1645840209238647784661334604149803842698165587987142 n 6 + 1134844313231332470158869524195173701437845610976000 n 5 + 610300867325002286054098421051825634802503655467312 n 4 + 249402353039405017564618652407486858607892258013776 n 3 + 74443653047975883377715939612216447160171847231904 n 2 + 15223444603275518010985388750086094191548167961600 n + 1895610363680724443833958583577668835547602060800 n + 107748879845369580406657989720932041087815168000) F(n + 2) + 78125 (10 n + 29) (10 n + 19) (5 n + 14) (10 n + 27) (10 n + 17) (5 n + 13) (2 n + 5) (5 n + 12) (10 n + 23) (10 n + 13) (5 n + 11) (10 n + 21) 12 (10 n + 11) (n + 3) (31631151990069989341341760 n 11 10 + 443845412702152021732041520 n + 2801648106593032849875436480 n 9 8 + 10503254709724877277083833800 n + 26000459079457858765957682196 n 7 6 + 44682385602007528156096219935 n + 54534111855520638477078814346 n 5 4 + 47500834298276763908439444629 n + 29217635690260039414602593162 n 3 2 + 12335143520029641052439234412 n + 3380442132771668110623929352 n + 537946450889369642333274528 n + 37482117176831185911847680) F(n + 3) = 0 and in Maple format -127401984*(6*n+17)*(6*n+11)*(6*n+5)*(3*n+5)*(3*n+2)*(2*n+3)*(2*n+1)*(3*n+4)*(3 *n+1)*(6*n+13)*(6*n+7)*(6*n+1)*(n+2)*(n+1)*(31631151990069989341341760*n^12+ 823419236582991893828142640*n^11+9771603677661324385456449360*n^10+ 69890086912088964626195669400*n^9+335495829594607960813857849996*n^8+ 1138521859146775520973744447423*n^7+2800226928403038874957054123859*n^6+ 5028595180854507257622282060236*n^5+6542531718883433282475330075342*n^4+ 6013469985835877898194794695771*n^3+3705745147593721221432173071323*n^2+ 1374471665328843740407971742290*n+232006820127400470851515693800)*F(n)-552960*( 6*n+17)*(6*n+11)*(3*n+5)*(2*n+3)*(3*n+4)*(6*n+13)*(6*n+7)*(n+2)*( 9587470239694542479580549597121190400*n^18+ 287930023598216908999507426488571747200*n^17+ 4021877375756635798330051427520713704000*n^16+ 34689677494238497057571063144228984647200*n^15+ 206851587187304227315977540614033360214800*n^14+ 904622068102959212353476594649472661237400*n^13+ 3003719974100166273302912427584923554247500*n^12+ 7733432024373755308471465432599999473960150*n^11+ 15630008693262114542137087593371584435138001*n^10+ 24946965741709714320490326701337465872205285*n^9+ 31467826499092483581505369869871230925508242*n^8+ 31237677151890886255508800106752972249110290*n^7+ 24172411400699527442985964446510965175946752*n^6+ 14349771599634571971675180709913315803026742*n^5+ 6374433012509013669326294664696915832278939*n^4+ 2038407166254294152478752122920339689752413*n^3+ 440375133986396777135117827809979390801206*n^2+ 57143812531234319584459751979012103157880*n+ 3339405646600612898553477526783997337600)*F(n+1)-3000*(10*n+19)*(6*n+17)*(10*n+ 17)*(10*n+13)*(6*n+13)*(10*n+11)*(62663602584859049144837469021671297399655680* n^20+2132561954240463724476820857712871596901568960*n^19+ 34082773774697012366585011375172505086093834560*n^18+ 339955581485702825273120443339315846765469445840*n^17+ 2372056594220852752944331289033277328223925784608*n^16+ 12299689181130742103419393973497841048072376747780*n^15+ 49142542460189050302672696923848522827303035982980*n^14+ 154804044204977372678319835669599317114483980302507*n^13+ 390149378455261315866151510206244436577194331589795*n^12+ 793690570437899682848898506496595829218322834548764*n^11+ 1309021009777480827729405852420047916482250598462710*n^10+ 1751269657988698314556992277015122217112241464501391*n^9+ 1894589397459230912118646639760883070265296031507891*n^8+ 1645840209238647784661334604149803842698165587987142*n^7+ 1134844313231332470158869524195173701437845610976000*n^6+ 610300867325002286054098421051825634802503655467312*n^5+ 249402353039405017564618652407486858607892258013776*n^4+ 74443653047975883377715939612216447160171847231904*n^3+ 15223444603275518010985388750086094191548167961600*n^2+ 1895610363680724443833958583577668835547602060800*n+ 107748879845369580406657989720932041087815168000)*F(n+2)+78125*(10*n+29)*(10*n+ 19)*(5*n+14)*(10*n+27)*(10*n+17)*(5*n+13)*(2*n+5)*(5*n+12)*(10*n+23)*(10*n+13)* (5*n+11)*(10*n+21)*(10*n+11)*(n+3)*(31631151990069989341341760*n^12+ 443845412702152021732041520*n^11+2801648106593032849875436480*n^10+ 10503254709724877277083833800*n^9+26000459079457858765957682196*n^8+ 44682385602007528156096219935*n^7+54534111855520638477078814346*n^6+ 47500834298276763908439444629*n^5+29217635690260039414602593162*n^4+ 12335143520029641052439234412*n^3+3380442132771668110623929352*n^2+ 537946450889369642333274528*n+37482117176831185911847680)*F(n+3) = 0 The initial conditions are as follows Pi 166263531392 Pi 164534670966016 E(0) = - ----, E(1) = - --------------- + ---------------, 20 125 39375 645239658416373636177408 Pi 127675644550636737131234459648 E(2) = - --------------------------- + ------------------------------ 3125 196828125 and in Maple notation E(0) = -1/20*Pi, E(1) = -166263531392/125*Pi+164534670966016/39375, E(2) = -\ 645239658416373636177408/3125*Pi+127675644550636737131234459648/196828125 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 7.3919707058929719080 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 -19073486328125 N + 4178824620233016799687500 N + 7636428363012249600 N + 3761479876608 and in Maple format -19073486328125*N^3+4178824620233016799687500*N^2+7636428363012249600*N+ 3761479876608 12 whose largest root let's call it a, is, 0.21909076024927279302 10 , and absolute value of the two smaller roots, let's call it b is, -6 0.94825367763146924691 10 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: /25\n /25\n C1(n) = d(10 n) |--| C(n), D1(n) = d(10 n) |--| D(n) \32/ \32/ are integers Defining /25\n E1(n) = d(10 n) |--| E(n) \32/ 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 /25\n n exp(10 n) |--| a \32/ and E1(n) is O of /25\n n exp(10 n) |--| b \32/ It follows that E1(n)=max(C1(n),D1(n))^(-delta) where delta equals 25 ln(b) + 10 + ln(--) 32 - ------------------- 25 ln(a) + 10 + ln(--) 32 12 recall that a is , 0.21909076024927279302 10 , and b is , -6 0.94825367763146924691 10 and delta is .11474701019856887780 implying the rigorous, but crude upper bound for the irrationality measure, \ 1+1/delta that equals 9.7148240138850430642 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 1.2203978066838219831 and the better delta is .15401430438715609921 that implies an irrationality measure, 1+1/delta that equals 7.4929033960782814603 ----------------------------------------- This ends this paper that took, 37.466, to generate.