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%I #41 Sep 29 2023 20:21:05
%S 5,2,2,2,2,0,0,7,10,2,0,4,0,0,4,2,16,2,2,0,0,2,0,8,2,2,1,4,0,2,2,0,2,
%T 0,2,8,6,2,0,2,2,0,2,4,0,0,0,2,2,2,0,2,0,2,2,2,6,0,0,0,0,0,4,5,8,0,0,
%U 4,0,0,2,2,12,0,0,2,0,0,2,8,2,2,0,0,0,0,0,0,8,0,2,2,0,2,0,0,2,2,2,12
%N Number of integral solutions to Mordell's equation y^2 = x^3 + n.
%C Mordell's equation has a finite number of integral solutions for all nonzero n.
%C Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
%C Sequence A054504 gives n for which there are no integral solutions. See A081120 for the number of integral solutions to y^2 = x^3 - n.
%C a(n) is odd iff n is a cube. - _Bernard Schott_, Nov 23 2019
%C From _Jianing Song_, Aug 24 2022: (Start)
%C a(n) = 5 if n is a sixth power. Further more, if A060950(n) = 0 (namely the elliptic curve y^2 = x^3 + n has rank 0), then:
%C - a(n) = 2 if n is a square but not a sixth power;
%C - a(n) = 1 if n is a cube but not a sixth power;
%C - a(n) = 0 otherwise.
%C This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
%C - If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
%C - If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
%C - If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
%C - If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
%C - In all the other cases, the torsion group is trivial.
%C So a torsion point on y^2 = x^3 + n other than O is an integral point. If y^2 = x^3 + n has rank 0, then all the integral points on y^2 = x^3 + n are exactly the torsion points other than O.
%C Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060950(n) = 0: the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, so it has the same Mordell-Weil group (hence the same rank and isomorphic torsion group) as y^2 = x^3 + n. (End)
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
%D J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.
%H Jean-François Alcover, <a href="/A081119/b081119.txt">Table of n, a(n) for n = 1..10000</a> [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and was based on the work of J. Gebel.]
%H Michael A. Bennett, Amir Ghadermarzi <a href="http://www.math.ubc.ca/~bennett/BeGh-LMSJCM-2015.pdf">Mordell's equation: a classical approach</a>. LMS J. Compute. Math. 18 (2015): 633-646. <a href="https://doi.org/10.1112%2FS1461157015000182">doi:10.1112/S1461157015000182</a> <a href="https://arxiv.org/abs/1311.7077">arXiv:1311.7077</a>
%H J. Gebel, A. Pethö, H. G. Zimmer, <a href="https://doi.org/10.1023%2FA%3A1000281602647">On Mordell's equation</a>, Compositio Mathematica. 110:3 (1998): 335-367.
%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a>. [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H Joseph H. Silverman, <a href="https://www.math.ens.psl.eu/~benoist/refs/Silverman.pdf">The Arithmetic of Elliptic Curves</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mordell_curve">Mordell curve</a>.
%t (* This naive approach gives correct results up to n = 1000 *) xmax[_] = 10^4; Do[xmax[n] = 10^5, {n, {297, 377, 427, 885, 899}}]; Do[xmax[n] = 10^6, {n, {225, 353, 618 }}]; f[n_] := (x = -Ceiling[n^(1/3)]-1; s = {}; While[x <= xmax[n], x++; y2 = x^3 + n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n]; If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0] ]); Table[an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* _Jean-François Alcover_, Oct 18 2011 *)
%Y Cf. A054504, A081120. See A134108 for another version.
%K nice,nonn
%O 1,1
%A _T. D. Noe_, Mar 06 2003
%E Edited by _Max Alekseyev_, Feb 06 2021
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