A179163 Numbers k such that Mordell's equation y^2 = x^3 - k has exactly 1 integral solution.

1, 8, 27, 64, 125, 512, 729, 1000, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 15625, 19683, 24389, 27000, 32768, 35937, 39304, 42875, 46656, 50653, 59319, 64000, 68921, 79507, 91125, 97336, 110592, 117649, 125000, 132651

Offset

1,2

Comments

Contains all sixth powers: suppose that y^2 = x^3 - t^6, then (y/t^3)^2 = (x/t^2)^3 - 1. The elliptic curve Y^2 = X^3 - 1 has rank 0 and the only rational points on it are (1,0), so y^2 = x^3 - t^6 has only one solution (t^2,0). - Jianing Song, Aug 24 2022

Links

Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Formula

a(n) = A356713(n)^3. - Jianing Song, Aug 24 2022

Mathematica

(* Assuming every term is a cube *) xmax = 2000; r[n_] := Reap[Do[rpos = Reduce[y^2 == x^3 - n, y, Integers]; If[rpos =!= False, Sow[rpos]]; rneg = Reduce[y^2 == (-x)^3 - n, y, Integers]; If[rneg =!= False, Sow[rneg]], {x, 1, xmax}]]; ok[1] = True; ok[n_] := Which[rn = r[n]; rn[[2]] === {}, False, Length[rn[[2]]] > 1, False, ! FreeQ[rn[[2, 1]], Or], False, True, True]; ok[n_ /; !IntegerQ[n^(1/3)]] = False; A179163 = Reap[Do[If[ok[n], Print[n]; Sow[n]], {n, 1, 140000}]][[2, 1]] (* Jean-François Alcover, Apr 12 2012 *)

Crossrefs

Cf. A081120, A081121, A179163-A179175, A356713.

Complement of A179149 among the positive cubes.

Cf. also A179145, A356703.

Sequence in context: A111131 A111103 A076969 * A050462 A112662 A121652

Adjacent sequences: A179160 A179161 A179162 * A179164 A179165 A179166

Keyword

nonn

Author

Artur Jasinski, Jun 30 2010

Extensions

Edited and extended by Ray Chandler, Jul 11 2010

Status

approved