Doing a=, 1, i.e. , ln(2) The best choice of the Rukhazde integral By Shalosh B. Ekhad consider the integral / A B\n |x (1 - x) | |-----------| | C | \ (1 + x) / -------------- 1 + x When A=B=C=1 it is the Alladi-Robinson integral. Can we do better? 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 ea\ ch case, the smallest 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-Ze\ ilberger 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 that exce\ ed the Alladi-Robinson choice of A=1,B=1,C=1 Our base line is the value for A=1,B=1,C=1 that is, 0.28203665564426164570 (A,B,C)=, 3, 3, 2, EmpDel=, 0.29878659722014938624 (A,B,C)=, 2, 4, 3, EmpDel=, 0.29878659722014938624 (A,B,C)=, 4, 2, 3, EmpDel=, 0.29878659722014938624 (A,B,C)=, 4, 4, 3, EmpDel=, 0.32076293736237401400 (A,B,C)=, 3, 3, 4, EmpDel=, 0.29878659722014938624 (A,B,C)=, 3, 5, 4, EmpDel=, 0.32076293736237401400 (A,B,C)=, 5, 3, 4, EmpDel=, 0.32076293736237401400 (A,B,C)=, 5, 5, 4, EmpDel=, 0.33989156046302040704 (A,B,C)=, 4, 4, 5, EmpDel=, 0.32076293736237401400 (A,B,C)=, 4, 6, 5, EmpDel=, 0.33989156046302040704 (A,B,C)=, 6, 4, 5, EmpDel=, 0.33989156046302040704 (A,B,C)=, 6, 6, 5, EmpDel=, 0.34284495455751294368 (A,B,C)=, 7, 7, 5, EmpDel=, 0.31081563362928378503 (A,B,C)=, 5, 5, 6, EmpDel=, 0.33989156046302040704 (A,B,C)=, 5, 7, 6, EmpDel=, 0.34284495455751294368 (A,B,C)=, 7, 5, 6, EmpDel=, 0.34284495455751294368 (A,B,C)=, 7, 7, 6, EmpDel=, 0.34169548801434955133 (A,B,C)=, 8, 7, 6, EmpDel=, 0.28251001108006587283 (A,B,C)=, 5, 9, 7, EmpDel=, 0.31081563362928378503 (A,B,C)=, 6, 6, 7, EmpDel=, 0.34284495455751294368 (A,B,C)=, 6, 8, 7, EmpDel=, 0.34169548801434955133 (A,B,C)=, 8, 6, 7, EmpDel=, 0.34169548801434955133 (A,B,C)=, 8, 8, 7, EmpDel=, 0.34541317922122995947 (A,B,C)=, 9, 5, 7, EmpDel=, 0.31081563362928378503 (A,B,C)=, 9, 6, 7, EmpDel=, 0.28251001108006587283 (A,B,C)=, 9, 8, 7, EmpDel=, 0.28229772462793471690 (A,B,C)=, 9, 9, 7, EmpDel=, 0.33357790582060898267 (A,B,C)=, 10, 10, 7, EmpDel=, 0.30653033978568514471 (A,B,C)=, 6, 9, 8, EmpDel=, 0.28251001108006587283 (A,B,C)=, 7, 7, 8, EmpDel=, 0.34169548801434955133 (A,B,C)=, 7, 9, 8, EmpDel=, 0.34541317922122995947 (A,B,C)=, 9, 7, 8, EmpDel=, 0.34541317922122995947 (A,B,C)=, 9, 9, 8, EmpDel=, 0.34288752753019938822 (A,B,C)=, 10, 7, 8, EmpDel=, 0.28229772462793471690 (A,B,C)=, 10, 9, 8, EmpDel=, 0.29023947191048410579 (A,B,C)=, 11, 11, 8, EmpDel=, 0.31291791298124560653 (A,B,C)=, 7, 7, 9, EmpDel=, 0.31081563362928378503 (A,B,C)=, 7, 8, 9, EmpDel=, 0.28251001108006587283 (A,B,C)=, 7, 10, 9, EmpDel=, 0.28229772462793471690 (A,B,C)=, 7, 11, 9, EmpDel=, 0.33357790582060898267 (A,B,C)=, 8, 8, 9, EmpDel=, 0.34541317922122995947 (A,B,C)=, 8, 10, 9, EmpDel=, 0.34288752753019938822 (A,B,C)=, 10, 8, 9, EmpDel=, 0.34288752753019938822 (A,B,C)=, 10, 10, 9, EmpDel=, 0.34377811703593302398 (A,B,C)=, 10, 11, 9, EmpDel=, 0.28501237172485662195 (A,B,C)=, 11, 7, 9, EmpDel=, 0.33357790582060898267 (A,B,C)=, 11, 8, 9, EmpDel=, 0.29023947191048410579 (A,B,C)=, 11, 10, 9, EmpDel=, 0.29740112854249922393 (A,B,C)=, 11, 11, 9, EmpDel=, 0.34072981988846617531 (A,B,C)=, 12, 11, 9, EmpDel=, 0.28963122903187968721 (A,B,C)=, 7, 13, 10, EmpDel=, 0.30653033978568514471 (A,B,C)=, 8, 9, 10, EmpDel=, 0.28229772462793471690 (A,B,C)=, 8, 11, 10, EmpDel=, 0.29023947191048410579 (A,B,C)=, 9, 9, 10, EmpDel=, 0.34288752753019938822 (A,B,C)=, 9, 11, 10, EmpDel=, 0.34377811703593302398 (A,B,C)=, 9, 12, 10, EmpDel=, 0.28501237172485662195 (A,B,C)=, 11, 9, 10, EmpDel=, 0.34377811703593302398 (A,B,C)=, 11, 11, 10, EmpDel=, 0.34184643588096967441 (A,B,C)=, 12, 9, 10, EmpDel=, 0.29740112854249922393 (A,B,C)=, 12, 11, 10, EmpDel=, 0.29603579620100444855 (A,B,C)=, 12, 13, 10, EmpDel=, 0.28358923119478164325 (A,B,C)=, 13, 7, 10, EmpDel=, 0.30653033978568514471 (A,B,C)=, 13, 12, 10, EmpDel=, 0.29367223718879648419 (A,B,C)=, 13, 13, 10, EmpDel=, 0.32659272533646160500 (A,B,C)=, 8, 14, 11, EmpDel=, 0.31291791298124560653 (A,B,C)=, 9, 9, 11, EmpDel=, 0.33357790582060898267 (A,B,C)=, 9, 10, 11, EmpDel=, 0.29023947191048410579 (A,B,C)=, 9, 12, 11, EmpDel=, 0.29740112854249922393 (A,B,C)=, 9, 13, 11, EmpDel=, 0.34072981988846617531 (A,B,C)=, 10, 10, 11, EmpDel=, 0.34377811703593302398 (A,B,C)=, 10, 12, 11, EmpDel=, 0.34184643588096967441 (A,B,C)=, 12, 9, 11, EmpDel=, 0.28501237172485662195 (A,B,C)=, 12, 10, 11, EmpDel=, 0.34184643588096967441 (A,B,C)=, 12, 12, 11, EmpDel=, 0.33769525694098348976 (A,B,C)=, 12, 13, 11, EmpDel=, 0.28521109561156887699 (A,B,C)=, 13, 9, 11, EmpDel=, 0.34072981988846617531 (A,B,C)=, 13, 10, 11, EmpDel=, 0.29603579620100444855 (A,B,C)=, 13, 12, 11, EmpDel=, 0.29657815741944673375 (A,B,C)=, 13, 13, 11, EmpDel=, 0.34325498478417945823 (A,B,C)=, 13, 14, 11, EmpDel=, 0.28839944278025241807 (A,B,C)=, 14, 8, 11, EmpDel=, 0.31291791298124560653 (A,B,C)=, 14, 9, 11, EmpDel=, 0.28963122903187968721 (A,B,C)=, 14, 13, 11, EmpDel=, 0.29743593900267807385 (A,B,C)=, 14, 14, 11, EmpDel=, 0.33722056255286703662 (A,B,C)=, 9, 14, 12, EmpDel=, 0.28963122903187968721 (A,B,C)=, 10, 11, 12, EmpDel=, 0.29740112854249922393 (A,B,C)=, 10, 13, 12, EmpDel=, 0.29603579620100444855 (A,B,C)=, 10, 15, 12, EmpDel=, 0.28358923119478164325 (A,B,C)=, 11, 10, 12, EmpDel=, 0.28501237172485662195 (A,B,C)=, 11, 11, 12, EmpDel=, 0.34184643588096967441 (A,B,C)=, 11, 13, 12, EmpDel=, 0.33769525694098348976 (A,B,C)=, 11, 14, 12, EmpDel=, 0.28521109561156887699 (A,B,C)=, 13, 11, 12, EmpDel=, 0.33769525694098348976 (A,B,C)=, 13, 13, 12, EmpDel=, 0.33535861311597942236 (A,B,C)=, 13, 14, 12, EmpDel=, 0.28870557717320678593 (A,B,C)=, 14, 11, 12, EmpDel=, 0.29657815741944673375 (A,B,C)=, 14, 13, 12, EmpDel=, 0.30129182065705458979 (A,B,C)=, 14, 15, 12, EmpDel=, 0.29097534800389492967 (A,B,C)=, 15, 10, 12, EmpDel=, 0.29367223718879648419 (A,B,C)=, 15, 14, 12, EmpDel=, 0.30209668243967511294 (A,B,C)=, 10, 10, 13, EmpDel=, 0.30653033978568514471 (A,B,C)=, 10, 15, 13, EmpDel=, 0.29367223718879648419 (A,B,C)=, 10, 16, 13, EmpDel=, 0.32659272533646160500 (A,B,C)=, 11, 11, 13, EmpDel=, 0.34072981988846617531 (A,B,C)=, 11, 12, 13, EmpDel=, 0.29603579620100444855 (A,B,C)=, 11, 14, 13, EmpDel=, 0.29657815741944673375 (A,B,C)=, 11, 15, 13, EmpDel=, 0.34325498478417945823 (A,B,C)=, 11, 16, 13, EmpDel=, 0.28839944278025241807 (A,B,C)=, 12, 12, 13, EmpDel=, 0.33769525694098348976 (A,B,C)=, 12, 14, 13, EmpDel=, 0.33535861311597942236 (A,B,C)=, 12, 15, 13, EmpDel=, 0.28870557717320678593 (A,B,C)=, 14, 11, 13, EmpDel=, 0.28521109561156887699 (A,B,C)=, 14, 12, 13, EmpDel=, 0.33535861311597942236 (A,B,C)=, 14, 14, 13, EmpDel=, 0.33503562335654483311 (A,B,C)=, 14, 15, 13, EmpDel=, 0.28833514012276504131 (A,B,C)=, 15, 10, 13, EmpDel=, 0.28358923119478164325 (A,B,C)=, 15, 11, 13, EmpDel=, 0.34325498478417945823 (A,B,C)=, 15, 12, 13, EmpDel=, 0.30129182065705458979 (A,B,C)=, 15, 14, 13, EmpDel=, 0.30049770199989319768 (A,B,C)=, 15, 15, 13, EmpDel=, 0.34555605606866370885 (A,B,C)=, 15, 16, 13, EmpDel=, 0.29168054779693632846 (A,B,C)=, 16, 10, 13, EmpDel=, 0.32659272533646160500 (A,B,C)=, 16, 11, 13, EmpDel=, 0.29743593900267807385 (A,B,C)=, 16, 15, 13, EmpDel=, 0.30106355623308048530 (A,B,C)=, 16, 16, 13, EmpDel=, 0.34062010903212920274 (A,B,C)=, 11, 11, 14, EmpDel=, 0.31291791298124560653 (A,B,C)=, 11, 12, 14, EmpDel=, 0.28963122903187968721 (A,B,C)=, 11, 16, 14, EmpDel=, 0.29743593900267807385 (A,B,C)=, 11, 17, 14, EmpDel=, 0.33722056255286703662 (A,B,C)=, 12, 13, 14, EmpDel=, 0.29657815741944673375 (A,B,C)=, 12, 15, 14, EmpDel=, 0.30129182065705458979 (A,B,C)=, 12, 17, 14, EmpDel=, 0.29097534800389492967 (A,B,C)=, 13, 12, 14, EmpDel=, 0.28521109561156887699 (A,B,C)=, 13, 13, 14, EmpDel=, 0.33535861311597942236 (A,B,C)=, 13, 15, 14, EmpDel=, 0.33503562335654483311 (A,B,C)=, 13, 16, 14, EmpDel=, 0.28833514012276504131 (A,B,C)=, 15, 12, 14, EmpDel=, 0.28870557717320678593 (A,B,C)=, 15, 13, 14, EmpDel=, 0.33503562335654483311 (A,B,C)=, 15, 15, 14, EmpDel=, 0.33298026511119100922 (A,B,C)=, 15, 16, 14, EmpDel=, 0.29024881384435820510 (A,B,C)=, 16, 11, 14, EmpDel=, 0.28839944278025241807 (A,B,C)=, 16, 13, 14, EmpDel=, 0.30049770199989319768 (A,B,C)=, 16, 15, 14, EmpDel=, 0.30121434187350046866 (A,B,C)=, 16, 17, 14, EmpDel=, 0.29525398745041626503 (A,B,C)=, 17, 11, 14, EmpDel=, 0.33722056255286703662 (A,B,C)=, 17, 12, 14, EmpDel=, 0.30209668243967511294 (A,B,C)=, 17, 16, 14, EmpDel=, 0.30570143723196532197 (A,B,C)=, 17, 17, 14, EmpDel=, 0.34359096146244417282 (A,B,C)=, 12, 13, 15, EmpDel=, 0.29367223718879648419 (A,B,C)=, 12, 17, 15, EmpDel=, 0.30209668243967511294 (A,B,C)=, 13, 12, 15, EmpDel=, 0.28358923119478164325 (A,B,C)=, 13, 13, 15, EmpDel=, 0.34325498478417945823 (A,B,C)=, 13, 14, 15, EmpDel=, 0.30129182065705458979 (A,B,C)=, 13, 16, 15, EmpDel=, 0.30049770199989319768 (A,B,C)=, 13, 17, 15, EmpDel=, 0.34555605606866370885 (A,B,C)=, 13, 18, 15, EmpDel=, 0.29168054779693632846 (A,B,C)=, 14, 13, 15, EmpDel=, 0.28870557717320678593 (A,B,C)=, 14, 14, 15, EmpDel=, 0.33503562335654483311 (A,B,C)=, 14, 16, 15, EmpDel=, 0.33298026511119100922 (A,B,C)=, 14, 17, 15, EmpDel=, 0.29024881384435820510 (A,B,C)=, 16, 13, 15, EmpDel=, 0.28833514012276504131 (A,B,C)=, 16, 14, 15, EmpDel=, 0.33298026511119100922 (A,B,C)=, 16, 16, 15, EmpDel=, 0.33061378415021291537 (A,B,C)=, 16, 17, 15, EmpDel=, 0.29114664828545641138 (A,B,C)=, 17, 12, 15, EmpDel=, 0.29097534800389492967 (A,B,C)=, 17, 13, 15, EmpDel=, 0.34555605606866370885 (A,B,C)=, 17, 14, 15, EmpDel=, 0.30121434187350046866 (A,B,C)=, 17, 16, 15, EmpDel=, 0.30178383348573327668 (A,B,C)=, 17, 17, 15, EmpDel=, 0.34447059302866765591 (A,B,C)=, 17, 18, 15, EmpDel=, 0.30295699742279787785 (A,B,C)=, 18, 13, 15, EmpDel=, 0.30106355623308048530 (A,B,C)=, 18, 16, 15, EmpDel=, 0.28406211632846539313 (A,B,C)=, 18, 17, 15, EmpDel=, 0.31168611576318473453 The highest Empirical delta is , 0.34555605606866370885, achieved by the case A=, 15, and B= , 15, and C=, 13 this leads to an irrationality measure around, 3.8938864836485314586 Hence it is worthwhile to try to investigate this case of, [15, 15, 13] ----------------------------------------- This ends this paper that took, 1972.748, to generate. Doing a=, 2, i.e. , ln(3/2) The best choice of the Rukhazde integral By Shalosh B. Ekhad consider the integral / A B\n |x (1 - x) | |-----------| | C | \ (2 + x) / -------------- 2 + x When A=B=C=1 it is the Alladi-Robinson integral. Can we do better? 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 ea\ ch case, the smallest 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-Ze\ ilberger 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 that exce\ ed the Alladi-Robinson choice of A=1,B=1,C=1 Our base line is the value for A=1,B=1,C=1 that is, 0.39794805105038829150 (A,B,C)=, 5, 5, 4, EmpDel=, 0.40373638273644551767 (A,B,C)=, 4, 6, 5, EmpDel=, 0.40373638273644551767 (A,B,C)=, 6, 4, 5, EmpDel=, 0.40373638273644551767 (A,B,C)=, 6, 6, 5, EmpDel=, 0.41313038622334154899 (A,B,C)=, 5, 5, 6, EmpDel=, 0.40373638273644551767 (A,B,C)=, 5, 7, 6, EmpDel=, 0.41313038622334154899 (A,B,C)=, 7, 5, 6, EmpDel=, 0.41313038622334154899 (A,B,C)=, 7, 7, 6, EmpDel=, 0.41730391403573415848 (A,B,C)=, 6, 6, 7, EmpDel=, 0.41313038622334154899 (A,B,C)=, 6, 8, 7, EmpDel=, 0.41730391403573415848 (A,B,C)=, 8, 6, 7, EmpDel=, 0.41730391403573415848 (A,B,C)=, 8, 8, 7, EmpDel=, 0.42424145320027722505 (A,B,C)=, 7, 7, 8, EmpDel=, 0.41730391403573415848 (A,B,C)=, 7, 9, 8, EmpDel=, 0.42424145320027722505 (A,B,C)=, 9, 7, 8, EmpDel=, 0.42424145320027722505 (A,B,C)=, 9, 9, 8, EmpDel=, 0.42506014676041120480 (A,B,C)=, 8, 8, 9, EmpDel=, 0.42424145320027722505 (A,B,C)=, 8, 10, 9, EmpDel=, 0.42506014676041120480 (A,B,C)=, 10, 8, 9, EmpDel=, 0.42506014676041120480 (A,B,C)=, 10, 10, 9, EmpDel=, 0.42977488046084648170 (A,B,C)=, 11, 11, 9, EmpDel=, 0.40881149399810275870 (A,B,C)=, 9, 9, 10, EmpDel=, 0.42506014676041120480 (A,B,C)=, 9, 11, 10, EmpDel=, 0.42977488046084648170 (A,B,C)=, 11, 9, 10, EmpDel=, 0.42977488046084648170 (A,B,C)=, 11, 11, 10, EmpDel=, 0.42987883521714850421 (A,B,C)=, 9, 13, 11, EmpDel=, 0.40881149399810275870 (A,B,C)=, 10, 10, 11, EmpDel=, 0.42977488046084648170 (A,B,C)=, 10, 12, 11, EmpDel=, 0.42987883521714850421 (A,B,C)=, 12, 10, 11, EmpDel=, 0.42987883521714850421 (A,B,C)=, 12, 12, 11, EmpDel=, 0.42879820652200173633 (A,B,C)=, 13, 9, 11, EmpDel=, 0.40881149399810275870 (A,B,C)=, 13, 13, 11, EmpDel=, 0.41566840693262587142 (A,B,C)=, 11, 11, 12, EmpDel=, 0.42987883521714850421 (A,B,C)=, 11, 13, 12, EmpDel=, 0.42879820652200173633 (A,B,C)=, 13, 11, 12, EmpDel=, 0.42879820652200173633 (A,B,C)=, 13, 13, 12, EmpDel=, 0.42755788396512256592 (A,B,C)=, 11, 11, 13, EmpDel=, 0.40881149399810275870 (A,B,C)=, 11, 15, 13, EmpDel=, 0.41566840693262587142 (A,B,C)=, 12, 12, 13, EmpDel=, 0.42879820652200173633 (A,B,C)=, 12, 14, 13, EmpDel=, 0.42755788396512256592 (A,B,C)=, 14, 12, 13, EmpDel=, 0.42755788396512256592 (A,B,C)=, 14, 14, 13, EmpDel=, 0.42863429395033771677 (A,B,C)=, 15, 11, 13, EmpDel=, 0.41566840693262587142 (A,B,C)=, 15, 15, 13, EmpDel=, 0.42449308085983232374 (A,B,C)=, 16, 16, 13, EmpDel=, 0.40681939820601252424 (A,B,C)=, 13, 13, 14, EmpDel=, 0.42755788396512256592 (A,B,C)=, 13, 15, 14, EmpDel=, 0.42863429395033771677 (A,B,C)=, 15, 13, 14, EmpDel=, 0.42863429395033771677 (A,B,C)=, 15, 15, 14, EmpDel=, 0.42709706929369304856 (A,B,C)=, 17, 17, 14, EmpDel=, 0.40998355376004472800 (A,B,C)=, 13, 13, 15, EmpDel=, 0.41566840693262587142 (A,B,C)=, 13, 17, 15, EmpDel=, 0.42449308085983232374 (A,B,C)=, 14, 14, 15, EmpDel=, 0.42863429395033771677 (A,B,C)=, 14, 16, 15, EmpDel=, 0.42709706929369304856 (A,B,C)=, 16, 14, 15, EmpDel=, 0.42709706929369304856 (A,B,C)=, 16, 16, 15, EmpDel=, 0.42710168216888245764 (A,B,C)=, 17, 13, 15, EmpDel=, 0.42449308085983232374 (A,B,C)=, 17, 17, 15, EmpDel=, 0.42579413371225288026 The highest Empirical delta is , 0.42987883521714850421, achieved by the case A=, 11, and B= , 11, and C=, 10 this leads to an irrationality measure around, 3.3262368790379297024 Hence it is worthwhile to try to investigate this case of, [11, 11, 10] ----------------------------------------- This ends this paper that took, 2343.261, to generate. Doing a=, 3, i.e. , ln(4/3) The best choice of the Rukhazde integral By Shalosh B. Ekhad consider the integral / A B\n |x (1 - x) | |-----------| | C | \ (3 + x) / -------------- 3 + x When A=B=C=1 it is the Alladi-Robinson integral. Can we do better? 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 ea\ ch case, the smallest 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-Ze\ ilberger 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 that exce\ ed the Alladi-Robinson choice of A=1,B=1,C=1 Our base line is the value for A=1,B=1,C=1 that is, 0.45382362062885585650 (A,B,C)=, 7, 7, 6, EmpDel=, 0.45532667563451549887 (A,B,C)=, 6, 8, 7, EmpDel=, 0.45532667563451549887 (A,B,C)=, 8, 6, 7, EmpDel=, 0.45532667563451549887 (A,B,C)=, 8, 8, 7, EmpDel=, 0.46330345176386536180 (A,B,C)=, 7, 7, 8, EmpDel=, 0.45532667563451549887 (A,B,C)=, 7, 9, 8, EmpDel=, 0.46330345176386536180 (A,B,C)=, 9, 7, 8, EmpDel=, 0.46330345176386536180 (A,B,C)=, 9, 9, 8, EmpDel=, 0.46582660634698589274 (A,B,C)=, 8, 8, 9, EmpDel=, 0.46330345176386536180 (A,B,C)=, 8, 10, 9, EmpDel=, 0.46582660634698589274 (A,B,C)=, 10, 8, 9, EmpDel=, 0.46582660634698589274 (A,B,C)=, 10, 10, 9, EmpDel=, 0.47078902550595832504 (A,B,C)=, 9, 9, 10, EmpDel=, 0.46582660634698589274 (A,B,C)=, 9, 11, 10, EmpDel=, 0.47078902550595832504 (A,B,C)=, 11, 9, 10, EmpDel=, 0.47078902550595832504 (A,B,C)=, 11, 11, 10, EmpDel=, 0.47123400620929086431 (A,B,C)=, 10, 10, 11, EmpDel=, 0.47078902550595832504 (A,B,C)=, 10, 12, 11, EmpDel=, 0.47123400620929086431 (A,B,C)=, 12, 10, 11, EmpDel=, 0.47123400620929086431 (A,B,C)=, 12, 12, 11, EmpDel=, 0.47125509943330486422 (A,B,C)=, 11, 11, 12, EmpDel=, 0.47123400620929086431 (A,B,C)=, 11, 13, 12, EmpDel=, 0.47125509943330486422 (A,B,C)=, 13, 11, 12, EmpDel=, 0.47125509943330486422 (A,B,C)=, 13, 13, 12, EmpDel=, 0.47231813976824862876 (A,B,C)=, 12, 12, 13, EmpDel=, 0.47125509943330486422 (A,B,C)=, 12, 14, 13, EmpDel=, 0.47231813976824862876 (A,B,C)=, 14, 12, 13, EmpDel=, 0.47231813976824862876 (A,B,C)=, 14, 14, 13, EmpDel=, 0.47299970162840223173 (A,B,C)=, 15, 15, 13, EmpDel=, 0.45968312003458948524 (A,B,C)=, 13, 13, 14, EmpDel=, 0.47231813976824862876 (A,B,C)=, 13, 15, 14, EmpDel=, 0.47299970162840223173 (A,B,C)=, 15, 13, 14, EmpDel=, 0.47299970162840223173 (A,B,C)=, 15, 15, 14, EmpDel=, 0.47387424612729508015 (A,B,C)=, 13, 17, 15, EmpDel=, 0.45968312003458948524 (A,B,C)=, 14, 14, 15, EmpDel=, 0.47299970162840223173 (A,B,C)=, 14, 16, 15, EmpDel=, 0.47387424612729508015 (A,B,C)=, 16, 14, 15, EmpDel=, 0.47387424612729508015 (A,B,C)=, 16, 16, 15, EmpDel=, 0.47225922778261930866 (A,B,C)=, 17, 13, 15, EmpDel=, 0.45968312003458948524 (A,B,C)=, 17, 17, 15, EmpDel=, 0.46392522888913825357 The highest Empirical delta is , 0.47387424612729508015, achieved by the case A=, 15, and B= , 15, and C=, 14 this leads to an irrationality measure around, 3.1102645019695240224 Hence it is worthwhile to try to investigate this case of, [15, 15, 14] ----------------------------------------- This ends this paper that took, 2546.434, to generate. Doing a=, 4, i.e. , ln(5/4) The best choice of the Rukhazde integral By Shalosh B. Ekhad consider the integral / A B\n |x (1 - x) | |-----------| | C | \ (4 + x) / -------------- 4 + x When A=B=C=1 it is the Alladi-Robinson integral. Can we do better? 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 ea\ ch case, the smallest 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-Ze\ ilberger 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 that exce\ ed the Alladi-Robinson choice of A=1,B=1,C=1 Our base line is the value for A=1,B=1,C=1 that is, 0.49443922956460802041 (A,B,C)=, 10, 10, 9, EmpDel=, 0.49623404685997491159 (A,B,C)=, 9, 11, 10, EmpDel=, 0.49623404685997491159 (A,B,C)=, 11, 9, 10, EmpDel=, 0.49623404685997491159 (A,B,C)=, 11, 11, 10, EmpDel=, 0.49764027854114780283 (A,B,C)=, 10, 10, 11, EmpDel=, 0.49623404685997491159 (A,B,C)=, 10, 12, 11, EmpDel=, 0.49764027854114780283 (A,B,C)=, 12, 10, 11, EmpDel=, 0.49764027854114780283 (A,B,C)=, 12, 12, 11, EmpDel=, 0.49783222694491518404 (A,B,C)=, 11, 11, 12, EmpDel=, 0.49764027854114780283 (A,B,C)=, 11, 13, 12, EmpDel=, 0.49783222694491518404 (A,B,C)=, 13, 11, 12, EmpDel=, 0.49783222694491518404 (A,B,C)=, 13, 13, 12, EmpDel=, 0.49936493864462621692 (A,B,C)=, 12, 12, 13, EmpDel=, 0.49783222694491518404 (A,B,C)=, 12, 14, 13, EmpDel=, 0.49936493864462621692 (A,B,C)=, 14, 12, 13, EmpDel=, 0.49936493864462621692 (A,B,C)=, 14, 14, 13, EmpDel=, 0.50143456768771307861 (A,B,C)=, 13, 13, 14, EmpDel=, 0.49936493864462621692 (A,B,C)=, 13, 15, 14, EmpDel=, 0.50143456768771307861 (A,B,C)=, 15, 13, 14, EmpDel=, 0.50143456768771307861 (A,B,C)=, 15, 15, 14, EmpDel=, 0.50138414706988099489 (A,B,C)=, 14, 14, 15, EmpDel=, 0.50143456768771307861 (A,B,C)=, 14, 16, 15, EmpDel=, 0.50138414706988099489 (A,B,C)=, 16, 14, 15, EmpDel=, 0.50138414706988099489 (A,B,C)=, 16, 16, 15, EmpDel=, 0.50105909355421167303 The highest Empirical delta is , 0.50143456768771307861, achieved by the case A=, 14, and B= , 14, and C=, 13 this leads to an irrationality measure around, 2.9942781460228066738 Hence it is worthwhile to try to investigate this case of, [14, 14, 13] ----------------------------------------- This ends this paper that took, 2694.898, to generate. Doing a=, 5, i.e. , ln(6/5) The best choice of the Rukhazde integral By Shalosh B. Ekhad consider the integral / A B\n |x (1 - x) | |-----------| | C | \ (5 + x) / -------------- 5 + x When A=B=C=1 it is the Alladi-Robinson integral. Can we do better? 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 ea\ ch case, the smallest 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-Ze\ ilberger 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 that exce\ ed the Alladi-Robinson choice of A=1,B=1,C=1 Our base line is the value for A=1,B=1,C=1 that is, 0.52003714296489332089 (A,B,C)=, 14, 14, 13, EmpDel=, 0.52062412449014076393 (A,B,C)=, 13, 15, 14, EmpDel=, 0.52062412449014076393 (A,B,C)=, 15, 13, 14, EmpDel=, 0.52062412449014076393 (A,B,C)=, 15, 15, 14, EmpDel=, 0.52123258986689668965 (A,B,C)=, 14, 14, 15, EmpDel=, 0.52062412449014076393 (A,B,C)=, 14, 16, 15, EmpDel=, 0.52123258986689668965 (A,B,C)=, 16, 14, 15, EmpDel=, 0.52123258986689668965 (A,B,C)=, 16, 16, 15, EmpDel=, 0.52091435382074819066 The highest Empirical delta is , 0.52123258986689668965, achieved by the case A=, 15, and B= , 15, and C=, 14 this leads to an irrationality measure around, 2.9185293081066987809 Hence it is worthwhile to try to investigate this case of, [15, 15, 14] ----------------------------------------- This ends this paper that took, 3093.994, to generate.