

A081120


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


24



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
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OFFSET

1,2


COMMENTS

Mordell's equation has a finite number of integral solutions for all nonzero n. Gebel computes the solutions for n < 10^5. 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.


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, page 191.
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335367.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel, except for 704, which was corrected by JeanFrançois Alcover on Mar 06 2012)
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]
Eric Weisstein's World of Mathematics, Mordell Curve


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}] (* JeanFrançois Alcover, Mar 06 2012 *)


CROSSREFS

Cf. A081119, A081121.
Sequence in context: A123565 A258701 A246160 * A200038 A249093 A102392
Adjacent sequences: A081117 A081118 A081119 * A081121 A081122 A081123


KEYWORD

nice,nonn


AUTHOR

T. D. Noe, Mar 06 2003


STATUS

approved



