A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

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, 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, 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

Offset: 1

Author

T. D. Noe, Mar 06 2003

Keywords

• nice
• nonn

Comments

Mordell's equation has a finite number of integral solutions for all nonzero n.

Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.

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.

a(n) is odd iff n is a cube. - Bernard Schott, Nov 23 2019

From Jianing Song, Aug 24 2022: (Start)

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:

- a(n) = 2 if n is a square but not a sixth power;

- a(n) = 1 if n is a cube but not a sixth power;

- a(n) = 0 otherwise.

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):

- 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).

- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).

- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).

- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).

- In all the other cases, the torsion group is trivial.

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.

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)

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
• J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..10000 [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.]
• Michael A. Bennett, Amir Ghadermarzi Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
• J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
• J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
• Joseph H. Silverman, The Arithmetic of Elliptic Curves.
• Eric Weisstein's World of Mathematics, Mordell Curve.
• Wikipedia, Mordell curve.

Crossrefs

Cf. A054504, A081120. See A134108 for another version.

Programs

Mathematica

(* 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 *)

Extensions

Edited by Max Alekseyev, Feb 06 2021