--------------------------------------- Trying to Improve Shalikohov's Upper Bound for the Irrationality Measure of Pi By Shalosh B. Ekhad In V. Kh. Shalokhov's beautiful article "On the Measure of Irrationality of the Number Pi" Mathematical Notes,2010, vol.88, no. 4, 563-573 finds a sequence of rational approximations to Pi based on the integral 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 This lead him, after a fairly painful proof to rigorously prove an upper bound of 7.606308.. . This inspired us to search for variations where the exponents 2*3 in the numerator, and 2*5 in the denominator are replaced by other choices A,B, relatively prime between 1 and 10 as well as (A,B)= (8,13) in the following generalization 4 + 2 I / | (2 A n) (2 A n) (2 A n) | -I (x - 4 + 2 I) (x - 4 - 2 I) (x - 5) | / 4 - 2 I (2 A n) (2 A n) / (2 B n + 1) (x - 6 + 2 I) (x - 6 - 2 I) / (x / (2 B n + 1) (x - 10) ) dx Since a rigorous investigation (even by computer) for each case is painful, it makes sense to first find the empirical deltas in abs(Pi-an/bn)<=CONSTANT/bn^(1+delta) for fairly large n. We call this delta the empirical delta, and take, in each case, the smallest delta for n between, 290, and , 300 This will give us an indication of what values of A and B are promising and worth pursuing. Thanks to the recurrences produced, in each case, by the amazing Almkvist-Zeilberger algorithm, it is very easy to crank-out many terms of these integrals, much faster then a direct computation. For the sake of completeness we will list ALL the empirical deltas, even the negative ones, that do not prove irrationality at all. (A,B)=, 1, 1, EmpDel=, -0.14318732995836029420 (A,B)=, 1, 2, EmpDel=, -0.025439649890311547704 (A,B)=, 1, 3, EmpDel=, -0.37167499585094753252 (A,B)=, 1, 4, EmpDel=, -0.65800662309423893518 (A,B)=, 1, 5, EmpDel=, -0.82478601044806211094 (A,B)=, 1, 6, EmpDel=, -0.91517737024886414562 (A,B)=, 1, 7, EmpDel=, -0.96809207646189277846 (A,B)=, 1, 8, EmpDel=, -0.99996746515647788518 (A,B)=, 1, 9, EmpDel=, -1.0002427497269898983 (A,B)=, 1, 10, EmpDel=, -1.0002108282091258680 (A,B)=, 2, 1, EmpDel=, -0.29482547329494001208 (A,B)=, 2, 3, EmpDel=, 0.16605428729395818514 (A,B)=, 2, 5, EmpDel=, -0.18378095670633231351 (A,B)=, 2, 7, EmpDel=, -0.54243908698650627018 (A,B)=, 2, 9, EmpDel=, -0.75428210168427261320 (A,B)=, 3, 1, EmpDel=, -0.35421678864852967524 (A,B)=, 3, 2, EmpDel=, -0.23721092098954426375 (A,B)=, 3, 4, EmpDel=, 0.050451296740565359479 (A,B)=, 3, 5, EmpDel=, 0.15727140930557009691 (A,B)=, 3, 7, EmpDel=, -0.10763869415952211337 (A,B)=, 3, 8, EmpDel=, -0.25763812328173805981 (A,B)=, 3, 10, EmpDel=, -0.48761745774431617747 (A,B)=, 4, 1, EmpDel=, -0.38236768936976593815 (A,B)=, 4, 3, EmpDel=, -0.20748488923458448202 (A,B)=, 4, 5, EmpDel=, -0.0057884310436490388444 (A,B)=, 4, 7, EmpDel=, 0.10930500866720511174 (A,B)=, 4, 9, EmpDel=, -0.074958832465554157508 (A,B)=, 5, 1, EmpDel=, -0.39943133428941214165 (A,B)=, 5, 2, EmpDel=, -0.32929834967892903277 (A,B)=, 5, 3, EmpDel=, -0.25967616784937561035 (A,B)=, 5, 4, EmpDel=, -0.19386078858806835267 (A,B)=, 5, 6, EmpDel=, -0.033396153364424264483 (A,B)=, 5, 7, EmpDel=, 0.099959655208267251593 (A,B)=, 5, 8, EmpDel=, 0.15701995819256081077 (A,B)=, 5, 9, EmpDel=, 0.088895901297894801813 (A,B)=, 6, 1, EmpDel=, -0.41083660426846207827 (A,B)=, 6, 5, EmpDel=, -0.18268269769483919520 (A,B)=, 6, 7, EmpDel=, -0.052489829579037258511 (A,B)=, 7, 1, EmpDel=, -0.41922584272669349045 (A,B)=, 7, 2, EmpDel=, -0.36930712192561646339 (A,B)=, 7, 3, EmpDel=, -0.31889666595993864172 (A,B)=, 7, 4, EmpDel=, -0.27052243530019819108 (A,B)=, 7, 5, EmpDel=, -0.21685015058967809233 (A,B)=, 7, 6, EmpDel=, -0.17480292956391877915 (A,B)=, 7, 8, EmpDel=, -0.066146371492385138494 (A,B)=, 7, 9, EmpDel=, 0.017519557860280910922 (A,B)=, 7, 10, EmpDel=, 0.12451550531454231901 (A,B)=, 8, 1, EmpDel=, -0.42608202666122991904 (A,B)=, 8, 3, EmpDel=, -0.33834109598597087127 (A,B)=, 8, 5, EmpDel=, -0.25185497610093627743 (A,B)=, 8, 7, EmpDel=, -0.16971471314328840824 (A,B)=, 8, 9, EmpDel=, -0.076352445792310347813 (A,B)=, 9, 1, EmpDel=, -0.43115897876685812935 (A,B)=, 9, 2, EmpDel=, -0.39181922713387286163 (A,B)=, 9, 4, EmpDel=, -0.31393038160665173909 (A,B)=, 9, 5, EmpDel=, -0.27514560776045961385 (A,B)=, 9, 7, EmpDel=, -0.20107464486854735266 (A,B)=, 9, 8, EmpDel=, -0.16562181283830737908 (A,B)=, 9, 10, EmpDel=, -0.083413935126588803910 (A,B)=, 10, 1, EmpDel=, -0.43540625826966765621 (A,B)=, 10, 3, EmpDel=, -0.36407306053696699786 (A,B)=, 10, 7, EmpDel=, -0.22333306771480022409 (A,B)=, 10, 9, EmpDel=, -0.16114379767762466631 (A,B)=, 8, 13, EmpDel= , 0.15586354092162189848 The highest Empirical delta is , 0.16605428729395818514, achieved by the case A=, 2, and B= , 3 this leads to an irrationality measure around, 7.0221269579733674009 that is better than the empirical delta for (A,B)=(3,5) that is 0.15727140930557009691 Hence it is worthwhile to try to investigate this case of, [2, 3], in the hope of beating Salikhov's previous record. ----------------------------------------- This ends this paper that took, 547.420, to generate.