

A081119


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


38



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
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OFFSET

1,1


COMMENTS

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


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

JeanFrançois Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous bfile, which had 10000 terms contributed by T. D. Noe and based on the work of J. Gebel.]
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, {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}] (* JeanFrançois Alcover, Oct 18 2011 *)


CROSSREFS

Cf. A054504, A081120. See A134108 for another version.
Sequence in context: A129165 A190288 A325499 * A303579 A306966 A286016
Adjacent sequences: A081116 A081117 A081118 * A081120 A081121 A081122


KEYWORD

nice,nonn


AUTHOR

T. D. Noe, Mar 06 2003


STATUS

approved



