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%I #22 Jul 19 2018 04:01:37
%S -29,-25,-20,-14,-5,5,14,16,20,25,49,70,79,130,250,305,400,695,1555,
%T 1645,18895
%N Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.
%C This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points in this curve.
%C This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).
%H Roelof J. Stroeker and Benjamin M. M. de Weger, <a href="https://doi.org/10.1090/S0025-5718-99-01047-9">Elliptic binomial diophantine equations</a>, Math. Comp. 68 (1999), 1257-1281.
%e a(6) = 5: 5^3 - 525 * 5 + 10156 = 7656 = 88 * 87.
%Y Cf. A029728 (the complete list of solutions x to y^2=x^3+17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).
%K sign,fini,full
%O 1,1
%A _Tomohiro Yamada_, May 29 2018
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