

A081119


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


36



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.


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)
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, A081119. See A134108 for another version.
Sequence in context: A058841 A129165 A190288 * A303579 A286016 A119320
Adjacent sequences: A081116 A081117 A081118 * A081120 A081121 A081122


KEYWORD

nice,nonn


AUTHOR

T. D. Noe, Mar 06 2003


STATUS

approved



