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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 (list; graph; refs; listen; history; text; internal format)
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, Springer-Verlag, page 191.

J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

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}] (* Jean-Fran├žois 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

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Last modified December 17 04:31 EST 2018. Contains 318192 sequences. (Running on oeis4.)