A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

1, 37, 2, 2, 89, 7, 683, 18, 3, 19, 25469, 3, 3, 163, 137, 1853, 631, 3, 4, 449, 7, 11, 23417, 730511, 1872, 28747, 5, 11, 4, 4, 5353, 2538163, 15409, 53, 197, 17351, 5563, 13, 433, 2570129, 13, 1176498611, 53, 1241, 4, 25903, 15642626656646177, 14, 5, 592, 4033, 165889, 90, 181, 9109, 5266097, 5, 184223499139, 5, 5, 7, 52954777

Offset: 1

Author

Jean-François Alcover, May 11 2011

Keywords

• nonn

Comments

This sequence a(k) is computed so that equation a(k)^3 + y^3 = A020898(k)*z^3 holds.

The 4 sequences A020898 [i.e., n], A190356 [i.e., x], A190580 [i.e., y] and A190581 [i.e., z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3.

All x values above 25469 were obtained from Mishima's list and may not be the least positive solution.

Examples

			a(18) = 3 because A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.
		

Links

• Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
• Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
• Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Crossrefs

Cf. A020898, A060838, A190580, A190581

Programs

Mathematica

(* Let x = u + v and y = u - v *)
f[n_, m_] := (r =  Reduce[u > 0 && v > 0 && Mod[2*u^3 + 6*u*v^2, n] == 0, {u, v},  Integers] ;
uv={u,v}/.(ToRules/@ List@@ r[[All,-2;;-1]])/.C-> c;
xy = (s = {};
Do[sel =  Select[uv,  IntegerQ[((2*#1[[1]]^3 + 6*#1[[1]]*#1[[2]]^2)/n)^(1/ 3)] &];
If[sel =!= {}, AppendTo[s, sel] ], {c[1], 0, m}, {c[2], 0,  m}];
{#[[1]] + #[[2]], #[[1]] - #[[2]]} & /@ (s //
Flatten[#, 1] &)) // Select[#, Total[#] != 0 &] &;
nxyz =  xy /. {x_Integer, y_} -> {n, x, y, ((x^3 + y^3)/n)^(1/3)};
nxyz /. ({, x, y_, z_} /; {x, y, z} != {0, 0, 0} &&
GCD[x, y, z] != 1) :> (gd = GCD[x, y, z]; {n, x/gd, y/gd, z/gd})) // Union // Sort[#, #1[[2]] < #2[[2]] &] &;
g[n_] := (m0 = 1; While[(r = f[n, m0]) == {}, m0 = 2 m0];
r // First);
A020898 = {2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130}; km = Length[A020898]; (* xm(n) = some hard to compute values of x from Hisanori Mishima's list *) xm[22]=25469; xm[50]=23417; xm[51]=730511; xm[58]=28747; xm[68]=2538163; xm[69]=15409; xm[75]=17351; xm[85]=2570129; xm[87]=1176498611; xm[92]=25903; xm[94]=15642626656646177; xm[106]=165889; xm[114]=9109; xm[115]=5266097; xm[123]=184223499139; xm[130]=52954777; xm[n_] := xm[n] = g[n][[2]];
A190356 = Table[ n = A020898[[k]]; Print[xm[n]]; xm[n], {k, 1, km}] (* Jean-François Alcover, Jan 03 2012 *)

Extensions

Positions corresponding to n=124 and n=127 (which were not minimal) corrected by Jean-François Alcover

Extended to 62 terms by Jean-François Alcover, Jan 03 2012