A055735 - OEIS

%I #23 Oct 16 2017 20:19:54

%S -1,2,-2,18,-8,-19,3,-3,10,-7,4,-4,9,5,-5,-17,6,-6,19,7,-7,-15,8,-8,

%T 17,361,-26,9,-9,28,10,-10,11,-11,-23,-35,-71,12,-12,73,37,25,13,-13,

%U 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

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

%C 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.

%D 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.

%H Sean A. Irvine, <a href="/A055735/b055735.txt">Table of n, a(n) for n = 1..135</a>

%H H. C. Williams and R. Holte, <a href="http://dx.doi.org/10.1090/S0025-5718-1977-0434946-0">Computation of the solution of x^3 + D y^3 = 1</a>, Mathematics of Computation, Vol. 31, No. 139. (Jul., 1977), pp. 778-785.

%H H. C. Williams and C. R. Zarnke, <a href="/A005988/a005988.pdf">Computation of the solutions of the Diophantine equation x^3+dy^3=1</a>, Proc. Conf. Numerical Maths., Winnipeg (1971), 671-676. (Annotated scanned copy)

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

%t m = 400; s = {}; Do[ If[x*y < 0, r = Reduce[ d > 0 && x^3 + d*y^3 == 1, d, Integers];

%t If[r =!= False, AppendTo[s, d /. ToRules[r]]]], {x, -m, m}, {y, -m, m}]; dd = Union[s] [[1 ;; 77]];

%t 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 *)

%Y Cf. A005988, A259453 (y values).

%K nice,sign

%O 1,2

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