

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).


3



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
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OFFSET

1,2


COMMENTS

Theorem (NagellDelone): 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), 671676.


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. 778785.
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), 671676. (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 (* JeanFrançois 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



