A329922 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives y-values).

1, 1, 2, 2, 2, 0, 0, 3, 3, 3, 0, 2, 0, 0, 4, 4, 4, 19, 12, 0, 0, 7, 0, 5, 5, 5, 0, 6, 0, 83, 2, 0, 5, 0, 6, 6, 6, 37, 0, 16, 7, 0, 4, 6, 0, 0, 0, 7, 7, 7, 0, 5, 0, 9, 28, 8, 7, 0, 0, 0, 0, 0, 8, 8, 8, 0, 0, 2, 0, 0, 14, 8, 9, 0, 0, 7, 0, 0, 302, 9, 9, 9, 0, 0, 0, 0, 0, 0, 9, 0, 8, 10, 0, 11, 0, 0, 77, 21, 10, 10, 10, 0, 0, 0, 13, 59, 48, 10, 0, 0, 0, 29, 11, 0, 0, 0, 12, 0, 386, 11

Offset

1,3

Comments

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

References

See A081119.

Links

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

Example

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) = 19.

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][[2]];
a /@ Range[120]

Crossrefs

Cf. A054504, A081119 (number of solutions), A329921 (x-values).

Sequence in context: A034852 A212438 A112790 * A179835 A110857 A108867

Adjacent sequences: A329919 A329920 A329921 * A329923 A329924 A329925

Keyword

nonn

Author

Jean-François Alcover, Nov 24 2019

Status

approved