A134108 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.

3, 1, 1, 1, 1, 0, 0, 4, 5, 1, 0, 2, 0, 0, 2, 1, 8, 1, 1, 0, 0, 1, 0, 4, 1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 3, 0, 0, 0, 0, 0, 2, 3, 4, 0, 0, 2, 0, 0, 1, 1, 6, 0, 0, 1, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 6, 2, 0, 0, 0, 1

Offset: 1

Author

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

Keywords

• nonn

Comments

a(n) = A081119(n)/2 if A081119(n) is even, (A081119(n)+1)/2 if A081119(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 (this entry) and A134109 dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

Examples

			y^2 = x^3 + 1 has solutions (y, x) = (0, -1), (1, 0) and (3, 2), hence a(1) = 3.
y^2 = x^3 + 6 has no solutions, hence a(6) = 0.
y^2 = x^3 + 17 has 8 solutions (see A029727, A029728), hence a(17) = 8.
y^2 = x^3 + 27 has solution (y, x) = (0, -3), hence a(27) = 1.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..10000
• 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

Crossrefs

Cf. A081119, A029727, A029728, A134109.

Programs

Magma

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, n])) }: n in [1..122] ];

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_] := a[n] = (fn = f[n]; an = If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0]]; If[EvenQ[an], an/2, (an + 1)/2]); Table[ Print["a[", n, "] = ", a[n] ]; a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 20 2012 *)
A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081119[[n]]}, If[EvenQ[an], an/2, (an + 1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 24 2019 *)