This is a collection of candidates for Apery-style proofs using generalizing\ the Beukers Zeta(2) integral Each entry is a list of length 7 where The first entry is the definition of the constant as a 3F2 The second entry is its value to 100 decimals The third entry is the second-order linear recurrence operator in n and N a\ nnihilating Both sequence of rational numbers A(n) and B(n) such that that A(n)/B(n) converges to the constant The fourth entry is the pair [[A(0),A[1]],[B(0),B(1)]], of initial condition\ s enabling the fast computations of A(n) and B(n) using the recurrence The fifth entry is the estimate for the limit of log(E(n))/n as n goes to in\ finity of the INTEGERating factor, i.e. the sequence of rational numbers such that A(n)*E(n) and B(n)*E(n) are BOTH integers The sixth entry is the estimate for the implied irrationality measure. The seventh entry is the quadruple [a1,a2,b1,b2] such that the constant is I\ nt(Int(x^(-a1)*(1-x)^(-a2)*y^(-b1)*(1-y)^(-b2)/(1-x*y),x=0..1),y=0..1)/(\ Beta(-a1+1,-b1+1)*B(-a2+1,-b2+1)). If Maple can identify the constant, we will mention it. We found lots of cases with denominator 2 but they are all (conjecturally) l\ inear-fractionally equaivalent to log(2). Here is one of them [3F2(1,1,1/2;2,3/2;1), 1.3862943611198906188344642429163531361510002687205105\ 08241360018986787243939389431211726653992837375, 2 3 2 4 (1 + n) (1 + 2 n) (10 n + 17) 2 (220 n + 814 n + 950 n + 341) N - -------------------------------- + ----------------------------------- 2 (10 n + 7) (2 n + 5) (3 + 2 n) (10 n + 7) (2 n + 5) (3 + 2 n) 2 + N , [[0, -14/3], [1, -10/3]], 1.994230405, 11.68475635, [0, 0, 1/2, 0]] The value of the constant is , 2 ln(2) We found lots of cases with denominator 3 but they are all (conjecturally) l\ inear-fractionally equaivalent to sqrt(3)*Pi. Here is one of them [3F2(1,1,2/3;2,7/3;1), 1.2551974569368714023763130305686229291362649923709623\ 02279539741552492724505415534736499952848240675, 2 (15 n + 34) (2 + 3 n) (1 + n) (5 + 3 n) - ---------------------------------------- 2 (3 n + 4) (15 n + 19) (7 + 3 n) (2 + n) 3 2 3 (5 + 3 n) (165 n + 869 n + 1484 n + 820) N 2 + ---------------------------------------------- + N , 2 (15 n + 19) (7 + 3 n) (2 + n) [[0, -19/4], [1, -15/4]], 2.088344954, 15.14606111, [0, 0, 1/3, -2/3]] 1/2 4 Pi 3 The value of the constant is probably, -6 + --------- 3 We found lots of cases with denominator 4 but they are all (conjecturally) l\ inear-fractionally equaivalent to sqrt(2). Here is one of them [3F2(1,7/4,5/4;7/2,3;1), 1.35911464500822166215487559845513874255733339766580\ 4488820008617085009657532934630466139191917495080, 2 2 (4 n + 11) (20 n + 113 n + 158) (5 + 4 n) (5 + 2 n) (7 + 4 n) - --------------------------------------------------------------------- 2 16 (20 n + 73 n + 65) (4 n + 13) (7 + 2 n) (4 n + 9) (3 + n) (2 + n) 4 3 2 (4 n + 11) (3520 n + 33968 n + 120684 n + 186571 n + 105455) N 2 + ----------------------------------------------------------------- + N , 2 8 (20 n + 73 n + 65) (4 n + 13) (7 + 2 n) (3 + n) -133 [[0, -91/9], [1, ----]], 2.257934239, 32.48689926, [-3/4, -3/4, -1/4, -3/4] 18 ] 1/2 48 + 64 2 The value of the constant is probably, - ------------- 1/2 -51 - 36 2 We found lots of cases with denominator 5 but they are all (conjecturally) l\ inear-fractionally equaivalent to sqrt(5). Here is one of them [3F2(1,9/5,7/5;18/5,3;1), 1.4212207343351299442391746969711196356172306763517\ 51893024269442411258818833845513982731588477472963, 2 (5 n + 14) (25 n + 140 n + 194) (7 + 5 n) (12 + 5 n) (9 + 5 n) (8 + 5 n) - ------------------------------------------------------------------------- 2 25 (25 n + 90 n + 79) (5 n + 16) (18 + 5 n) (5 n + 11) (3 + n) (2 + n) 2 (5 n + 14) (5 n + 13) (12 + 5 n) (275 n + 1265 n + 1289) N 2 + ----------------------------------------------------------- + N , 2 5 (25 n + 90 n + 79) (5 n + 16) (18 + 5 n) (3 + n) -237 -414 [[0, ----], [1, ----]], 0.005733990166, 2.004777678, 22 55 [-4/5, -4/5, -2/5, -3/5]] 1/2 130 - 195 5 The value of the constant is probably, - -------------- 1/2 108 + 48 5 We found two cases of denominator 6 where we could identify the constant One of those, that can be expressed in terms of sqrt(3) is [3F2(1,11/6,3/2;11/3,3;1), 1.465908868652243591939817706973455809437345508850\ 249619679443346099719514607426738452372607912006323, (2 n + 5) (15 n + 46) (6 n + 17) (7 + 3 n) (11 + 6 n) (3 + 2 n) - --------------------------------------------------------------- 4 (15 n + 31) (6 n + 19) (11 + 3 n) (6 n + 13) (3 + n) (2 + n) 2 3 (6 n + 17) (2 n + 5) (330 n + 1837 n + 2531) N 2 + ------------------------------------------------- + N , 4 (15 n + 31) (6 n + 19) (11 + 3 n) (3 + n) -1023 -99 [[0, -----], [1, ---]], 2.287449767, 40.57115165, [-5/6, -5/6, -1/2, -1/2]] 91 13 The value of the constant is probably, FAIL ------------------------ Another one, that can be expressed in terms of the cubic root of 2, is [3F2(1,11/6,4/3;11/3,3;1), 1.390764502617977891731576274536725820106302943731\ 425106730752098822088693348207284376995058175754045, 2 (6 n + 17) (15 n + 85 n + 119) (4 + 3 n) (5 + 2 n) (11 + 6 n) (5 + 3 n) - ------------------------------------------------------------------------ 2 12 (15 n + 55 n + 49) (3 n + 10) (11 + 3 n) (6 n + 13) (3 + n) (2 + n) 2 2 (6 n + 17) (3 n + 8) (165 n + 770 n + 794) (7 + 3 n) N 2 + -------------------------------------------------------------- + N , 2 3 (3 + n) (6 n + 13) (11 + 3 n) (3 n + 10) (15 n + 55 n + 49) [[0, -11], [1, -55/7]], 2.183355822, 21.60775429, [-5/6, -5/6, -1/3, -2/3]] (1/3) 2/3 (-1944 + 1536 2 ) 2 The value of the constant is probably, - -------------------------- 10 The two classes that are not identified are coming up later. ------------------------ We found two cases of denominator 7 where we could identify the constant One of those, is a cubic irrationality is [3F2(1,13/7,11/7;26/7,3;1), 1.49959644167244534212297390018843200436204018359\ 4141589008443582709374560111390760289633360688430382, - (7 n + 20) 2 / (35 n + 194 n + 266) (11 + 7 n) (16 + 7 n) (13 + 7 n) (10 + 7 n) / (49 / 2 (35 n + 124 n + 107) (7 n + 22) (26 + 7 n) (7 n + 15) (3 + n) (2 + n)) 4 3 2 (7 n + 20) (18865 n + 180026 n + 632226 n + 964555 n + 535898) N 2 + ------------------------------------------------------------------- + N 2 7 (35 n + 124 n + 107) (7 n + 22) (26 + 7 n) (3 + n) -1391 -806 , [[0, -----], [1, ----]], 2.142992038, 18.29236149, 120 105 [-6/7, -6/7, -4/7, -3/7]] The constant seems to be, 1.4995964416724453421229739001884320043620401835941\ 41589008443582709374560111390760289633360688430382 Another one is also a cubic irrationality is [3F2(1,13/7,3/7;3,8/7;1), 1.7005662520620103928461373981715763262977302230652\ 27299441254449755468833406746841633841763727811239, - (5 + 7 n) (6 + 7 n) 3 2 / (3 + 7 n) (13 + 7 n) (245 n + 1470 n + 2872 n + 1830) / (49 (2 + n) / 3 2 (7 n + 2) (3 + n) (7 n + 25) (245 n + 735 n + 667 n + 183)) + (13 + 7 n) 5 4 3 2 (132055 n + 924385 n + 2425598 n + 2981111 n + 1703145 n + 356850) N / 3 2 2 / (7 (245 n + 735 n + 667 n + 183) (7 n + 25) (3 + n) (7 n + 2)) + N , / 61 43 [[0, --], [1, --]], 2.137331008, 17.90701436, [-6/7, -1/7, 4/7, 2/7]] 30 35 The constant seems to be, 1/2 1/3 245 (-28 + 84 I 3 ) 1715 203 ------------------------ + ------------------------ - --- 792 1/2 1/3 198 198 (-28 + 84 I 3 ) ----------------------------------------------- For the following case, Maple was unable to identify the constant, so there\ is hope that we have an irrationality proof of a constant not yet proved to be irrational. Two cases with denominator 6 [3F2(1,1,5/6;5/2,7/3;1), 1.23927789585817670115034342290529443833439969763174\ 9731275891767373125787672459329275340796730446125, 2 9 (15 n + 69 n + 79) (1 + n) (5 + 6 n) (3 + 2 n) - ------------------------------------------------- 2 2 4 (15 n + 39 n + 25) (3 n + 8) (7 + 3 n) 5 4 3 2 3 (1485 n + 13266 n + 46605 n + 80424 n + 68120 n + 22640) N 2 + ---------------------------------------------------------------- + N , 2 2 (15 n + 39 n + 25) (3 n + 8) (7 + 3 n) [[0, -45/8], [1, -9/2]], 2.393043644, 369.7226596, [0, -1/2, 1/6, -1/2]] [3F2(1,5/3,1/2;19/6,2;1), 1.2238415353141259357465361653564049871494980513489\ 88008552871978146719404715947523105712107717285377, 9 (3 + 2 n) (1 + 2 n) (5 + 3 n) (5 n + 14) (2 n + 5) - ---------------------------------------------------- 4 (5 n + 9) (n + 1) (3 n + 11) (19 + 6 n) (3 n + 4) 4 3 2 3 (2 n + 5) (990 n + 8382 n + 26189 n + 35787 n + 18044) N 2 + ------------------------------------------------------------- + N , 2 (5 n + 9) (3 n + 11) (19 + 6 n) (3 n + 4) (2 + n) -243 [[0, ----], [1, -99/8]], 2.388811189, 278.9967594, [-2/3, -1/2, 1/2, -1/2]] 16 Six cases with denominator 7 [3F2(1,13/7,11/7;26/7,23/7;1), 1.42657928503217978230037392797065258071377463\ 8570483371748262280117926376260343134781978342383006964, - (7 n + 20) 2 / (245 n + 1442 n + 2102) (11 + 7 n) (18 + 7 n) (13 + 7 n) (12 + 7 n) / (7 / 2 (245 n + 952 n + 905) (7 n + 22) (26 + 7 n) (7 n + 17) (23 + 7 n) (2 + n)) + (7 n + 20) (7 n + 19) (18 + 7 n) 4 3 2 / (18865 n + 175714 n + 601734 n + 899615 n + 495770) N / ( / 2 (245 n + 952 n + 905) (7 n + 22) (26 + 7 n) (7 n + 17) (23 + 7 n) (2 + n)) 2 -2353 + N , [[0, -----], [1, -39/5]], 2.291494165, 42.00340357, 210 [-6/7, -6/7, -4/7, -5/7]] [3F2(1,13/7,10/7;25/7,22/7;1), 1.42697139777377675063093466377867693738018795\ 1335212720916933002885220386740427811167251661393373009, (12 + 7 n) (17 + 7 n) (10 + 7 n) (13 + 7 n) (7 n + 19) (35 n + 92) - ------------------------------------------------------------------ 7 (7 n + 15) (35 n + 57) (25 + 7 n) (7 n + 16) (22 + 7 n) (2 + n) 3 2 11 (7 n + 19) (17 + 7 n) (245 n + 1589 n + 3366 n + 2328) N 2 + ------------------------------------------------------------- + N , (35 n + 57) (25 + 7 n) (7 n + 16) (22 + 7 n) (2 + n) [[0, -76/7], [1, -68/9]], 2.293532209, 42.76415297, [-6/7, -5/7, -3/7, -5/7]] [3F2(1,13/7,9/7;25/7,17/7;1), 1.575547900611281463616125774063343008932714694\ 262283895936599062393874733304208660291195520489064396, - (7 n + 20) 2 / (245 n + 1190 n + 1433) (9 + 7 n) (13 + 7 n) (12 + 7 n) (8 + 7 n) / (7 / 2 (245 n + 700 n + 488) (7 n + 23) (25 + 7 n) (7 n + 11) (17 + 7 n) (2 + n)) + (7 n + 20) (7 n + 16) 5 4 3 2 (132055 n + 1263955 n + 4757067 n + 8792945 n + 7975082 n + 2836344) N / 2 / ((245 n + 700 n + 488) (7 n + 23) (25 + 7 n) (7 n + 11) (17 + 7 n) / 2 -793 -143 (2 + n)) + N , [[0, ----], [1, ----]], 2.264826398, 34.07226039, 56 16 [-6/7, -5/7, -2/7, -1/7]] [3F2(1,13/7,8/7;24/7,16/7;1), 1.580356320266976855126243717022663241139673115\ 202621673814374156283492162577338646864791627083566077, - (7 n + 15) 2 / (245 n + 1302 n + 1724) (13 + 7 n) (12 + 7 n) (11 + 7 n) (8 + 7 n) / (7 / 2 (245 n + 812 n + 667) (7 n + 23) (24 + 7 n) (7 n + 10) (16 + 7 n) (2 + n)) + (7 n + 15) 5 4 3 2 (132055 n + 1362053 n + 5538225 n + 11100775 n + 10975340 n + 4284792) / 2 N / ((245 n + 812 n + 667) (7 n + 23) (24 + 7 n) (7 n + 10) (2 + n)) / 2 -667 + N , [[0, ----], [1, -10]], 2.280384590, 38.29032071, 42 [-6/7, -4/7, -1/7, -1/7]] [3F2(1,13/7,12/7;23/7,22/7;1), 1.67234841025118232998438522990354055817901745\ 8783924996678532906337269929987698604428785593092308753, (7 n + 20) (7 n + 17) (35 n + 94) (12 + 7 n) (13 + 7 n) (10 + 7 n) - ------------------------------------------------------------------ 7 (35 n + 59) (7 n + 18) (23 + 7 n) (7 n + 16) (22 + 7 n) (2 + n) 3 2 (7 n + 17) (7 n + 20) (2695 n + 17248 n + 36103 n + 24750) N 2 + -------------------------------------------------------------- + N , (35 n + 59) (7 n + 18) (23 + 7 n) (22 + 7 n) (2 + n) -7670 -650 [[0, -----], [1, ----]], 2.286954804, 40.40254971, [-6/7, -3/7, -5/7, -3/7] 693 99 ] [3F2(1,12/7,11/7;22/7,20/7;1), 1.66449355925700954891900725265452830534212014\ 3834685882646791099868077700479165154308612533573857248, (9 + 7 n) (10 + 7 n) (12 + 7 n) (7 n + 19) (35 n + 82) - ------------------------------------------------------ 7 (35 n + 47) (22 + 7 n) (7 n + 15) (20 + 7 n) (2 + n) 3 2 (7 n + 19) (2695 n + 15169 n + 27788 n + 16484) N 2 + --------------------------------------------------- + N , (35 n + 47) (22 + 7 n) (20 + 7 n) (2 + n) -141 [[0, ----], [1, -6]], 2.289974496, 41.45353517, [-5/7, -3/7, -4/7, -2/7]] 14 -----------------------