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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 (list; graph; refs; listen; history; text; internal format)
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, 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.]

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]

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

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}] (* Jean-François 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

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Last modified February 18 00:37 EST 2020. Contains 332006 sequences. (Running on oeis4.)