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

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, 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, 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

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

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.

From Jianing Song, Aug 24 2022: (Start)

If A060951(n) = 0 (namely the elliptic curve y^2 = x^3 - n has rank 0), then:

- a(n) = 2 if n is of the form 432*t^6;

- a(n) = 1 if n is a cube;

- 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 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)

Examples

			a(4)=4 refers to (x,y) = (2,+-2) and (5,+-11).
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

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 based on the work of J. Gebel.]
• M. A. Bennett and A. Ghadermarzi, Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
• J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica. 110:3 (1998): 335-367.
• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
• Joseph H. Silverman, The Arithmetic of Elliptic Curves.
• Eric Weisstein's World of Mathematics, Mordell Curve.
• Wikipedia, Mordell curve.

Crossrefs

Cf. A081119, A081121. See A134109 for another version.

Programs

Mathematica

(* 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 *)

Extensions

Edited by Max Alekseyev, Feb 06 2021