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A055735
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The x value of the unique nontrivial solution to x^3 + d*y^3 = 1 for all admissible d < 50000. d = 2,7,9,17,...,48949.
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0
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-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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| 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.
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EXAMPLE
| (-1)^3 + 2*1^3 = 1, 2^3 + 7*(-1)^3 = 1, etc...
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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 (* From Jean-François Alcover, Jun 8 2011 *)
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CROSSREFS
| Sequence in context: A000528 A074970 A087338 * A168296 A205454 A100304
Adjacent sequences: A055732 A055733 A055734 * A055736 A055737 A055738
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KEYWORD
| nice,sign
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AUTHOR
| Matt Herman (Henayni(AT)hotmail.com), Nov 28 2000
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