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

%I #8 Nov 27 2019 01:43:00

%S 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,

%T 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,

%U 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

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

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

%D See A081119.

%H Jean-François Alcover, <a href="/A329922/b329922.txt">Table of n, a(n) for n = 1..10000</a>

%e For n=12, the "min |x|" solution is 2^2 = (-2)^3+12, hence xy(12) = [-2,2] and a(12) = 2;

%e for n=18, it is 19^2 = 7^3 + 18, hence xy(18) = [7,19] and a(18) = 19.

%t A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {_, _}][[All, 2]];

%t r[n_, x_] := Reduce[y >= 0 && y^2 == x^3 + n, y, Integers];

%t 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[]]]];

%t a[n_] := xy[n][[2]];

%t a /@ Range[120]

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

%K nonn

%O 1,3

%A _Jean-François Alcover_, Nov 24 2019