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 A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n. 24

%I

%S 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,

%T 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,

%U 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

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

%C 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.

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

%H Jean-François Alcover, <a href="/A081120/b081120.txt">Table of n, a(n) for n = 1..10000</a> [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.]

%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

%H J. Gebel, A. Petho, G. Zimmer, <a href="https://doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Mathematica 110 (1998), 335-367.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>

%t (* 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 *)

%Y Cf. A081119, A081121.

%K nice,nonn

%O 1,2

%A _T. D. Noe_, Mar 06 2003

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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)