A055735 The x value of the unique nontrivial solution to x^3 + d*y^3 = 1 for all admissible (d = 2,7,9,17,..., A005988).

-1, 2, -2, 18, -8, -19, 3, -3, 10, -7, 4, -4, 9, 5, -5, -17, 6, -6, 19, 7, -7, -15, 8, -8, 17, 361, -26, 9, -9, 28, 10, -10, 11, -11, -23, -35, -71, 12, -12, 73, 37, 25, 13, -13, 14, -14, -44, 15, -15, 46, -31, -63, 16, -16, 65, 33, 17, -17, 361, -53, 18, -18, 55, 19, -19, -39, 20, -20, 41, -62, 21, -21, 64, 22, -22, 23, -23

Offset

1,2

Comments

Theorem (Nagell-Delone): The equation x^3 + d*y^3 = 1 has at most one nontrivial solution. If (e,f) is a solution, then e+f*d^(1/3) is either E or E^2, where E is the fundamental unit of Q adjoined with the cube root of d. The latter case occurs only for d = 19,20,28.

References

H. C. Williams and C. R. Zarnke, Computation of the solutions of the Diophantine equation x^3+dy^3=1, Proc. Conf. Numerical Maths., Winnipeg (1971), 671-676.

Links

Sean A. Irvine, Table of n, a(n) for n = 1..135

H. C. Williams and R. Holte, Computation of the solution of x^3 + D y^3 = 1, Mathematics of Computation, Vol. 31, No. 139. (Jul., 1977), pp. 778-785.

H. C. Williams and C. R. Zarnke, Computation of the solutions of the Diophantine equation x^3+dy^3=1, Proc. Conf. Numerical Maths., Winnipeg (1971), 671-676. (Annotated scanned copy)

Example

(-1)^3 + 2*1^3 = 1, 2^3 + 7*(-1)^3 = 1, etc...

Mathematica

m = 400; s = {}; Do[ If[x*y < 0, r = Reduce[ d > 0 && x^3 + d*y^3 == 1, d, Integers];
If[r =!= False, AppendTo[s, d /. ToRules[r]]]], {x, -m, m}, {y, -m, m}]; dd = Union[s] [[1 ;; 77]];
fi[d_] := x /. FindInstance[y != 0 && -m < x < m && x^3 + d*y^3 == 1, {x, y}, Integers] // First; fi /@ dd (* Jean-François Alcover, Jun 08 2011 *)

Crossrefs

Cf. A005988, A259453 (y values).

Sequence in context: A231123 A225123 A087338 * A168296 A205454 A100304

Adjacent sequences: A055732 A055733 A055734 * A055736 A055737 A055738

Keyword

nice,sign

Author

Matt Herman (Henayni(AT)hotmail.com), Nov 28 2000

Status

approved