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%I #4 Aug 23 2018 17:05:33
%S 31,74,76,87,112,121,319,562,679,1462,3937,5312,7987,18312,61312,
%T 66712,2597287
%N Complete list of solutions to y^2 + y = x^3 - 525x + 10156 with y > 0; sequence gives positive y 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 with y > 0 in this curve.
%C Each positive y corresponds to a negative solution -y - 1, so that the sequence gives all y values of solutions.
%C Some y values corresponds to three solutions. For y = 87, we have x = -25, 5 or 20. For y = 112, we have x = -20, -5 or 25. Any other value of y corresponds to a unique solution.
%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(1) = 31: (-29)^3 - 525 * (-29) + 10156 = 996 = 31 * 32.
%o (SageMath) EllipticCurve([0, 0, 1, -525, 10156]).integral_points()
%Y Cf. A303615 (x values)
%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 nonn,fini,full
%O 1,1
%A _Tomohiro Yamada_, Jul 20 2018
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