A329921 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives x-values).

0, -1, 1, 0, -1, 0, 0, 1, 0, -1, 0, -2, 0, 0, 1, 0, -1, 7, 5, 0, 0, 3, 0, 1, 0, -1, -3, 2, 0, 19, -3, 0, -2, 0, 1, 0, -1, 11, 0, 6, 2, 0, -3, -2, 0, 0, 0, 1, 0, -1, 0, -3, 0, 3, 9, 2, -2, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -4, 0, 0, 5, -2, 2, 0, 0, -3, 0, 0, 45, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, -3, 2, 0, 3

Offset: 1

Author

Jean-François Alcover, Nov 24 2019

Keywords

• sign

Comments

Conventionally, no solution is indicated by (x,y) = (0,0).

Examples

			For n=12, the "min |x|" solution is 2^2 = (-2)^3+12, hence xy(12) = [-2,2] and a(12) = -2;
for n=18, it is 19^2  = 7^3 + 18, hence xy(18) = [7,19] and a(18) = 7.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Mordell Curve

Crossrefs

Cf. A054504, A081119 (number of solutions), A134109, A329922 (y-values).

Programs

Mathematica

A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {, }][[All, 2]];
r[n_, x_] := Reduce[y >= 0 && y^2 == x^3 + n, y, Integers];
xy[n_] := If[A081119[[n]] == 0, {0, 0}, For[x = 0, True, x++, rn = r[n, x]; If[rn =!= False, Return[{x, y} /. ToRules[rn]]; Break[]]; rn = r[n, -x]; If[rn =!= False, Return[{-x, y} /. ToRules[rn]]; Break[]]]];
a[n_] := xy[n][[1]];
a /@ Range[120]