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A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.

%I A081120 #51 Feb 16 2025 08:32:48
%S A081120 1,2,0,4,0,0,4,1,0,0,4,0,2,0,2,0,0,2,2,2,0,0,2,0,2,4,1,6,0,0,0,0,0,0,
%T A081120 2,0,0,0,6,2,0,0,0,2,2,0,6,4,2,0,0,0,4,2,4,2,0,0,0,4,2,0,4,1,0,0,2,0,
%U A081120 0,0,2,2,0,2,0,4,0,0,2,0,2,0,2,0,0,0,2,0,2,0,0,0,0,0,2,0,0,0,0,6
%N A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.
%C A081120 Mordell's equation has a finite number of integral solutions for all nonzero n.
%C A081120 Gebel, Pethö, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
%C A081120 Sequence A081121 gives n for which there are no integral solutions. See A081119 for the number of integral solutions to y^2 = x^3 + n.
%C A081120 From _Jianing Song_, Aug 24 2022: (Start)
%C A081120 If A060951(n) = 0 (namely the elliptic curve y^2 = x^3 - n has rank 0), then:
%C A081120 - a(n) = 2 if n is of the form 432*t^6;
%C A081120 - a(n) = 1 if n is a cube;
%C A081120 - a(n) = 0 otherwise.
%C A081120 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 A081120 - 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 A081120 - If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
%C A081120 - If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
%C A081120 - If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
%C A081120 - In all the other cases, the torsion group is trivial.
%C A081120 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 A081120 Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060951(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 A081120 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
%H A081120 Jean-François Alcover, <a href="/A081120/b081120.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 based on the work of J. Gebel.]
%H A081120 M. A. Bennett and A. 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 A081120 J. Gebel, A. Pethö, and 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 A081120 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 A081120 Joseph H. Silverman, <a href="https://www.math.ens.psl.eu/~benoist/refs/Silverman.pdf">The Arithmetic of Elliptic Curves</a>.
%H A081120 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>.
%H A081120 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mordell_curve">Mordell curve</a>.
%e A081120 a(4)=4 refers to (x,y) = (2,+-2) and (5,+-11).
%t A081120 (* This naive approach gives correct results up to n=1000 *) xmax[_] = 10^4; Do[ xmax[n] = 10^5, {n, {366, 775, 999}}]; Do[ xmax[n] = 10^6, {n, {207, 307, 847}}]; f[n_] := (x = Floor[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_, Mar 06 2012 *)
%Y A081120 Cf. A081119, A081121. See A134109 for another version.
%K A081120 nice,nonn
%O A081120 1,2
%A A081120 _T. D. Noe_, Mar 06 2003
%E A081120 Edited by _Max Alekseyev_, Feb 06 2021