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

%I A179163 #26 Aug 24 2022 23:12:35
%S A179163 1,8,27,64,125,512,729,1000,1728,2197,2744,3375,4096,4913,5832,6859,
%T A179163 8000,9261,10648,15625,19683,24389,27000,32768,35937,39304,42875,
%U A179163 46656,50653,59319,64000,68921,79507,91125,97336,110592,117649,125000,132651
%N A179163 Numbers k such that Mordell's equation y^2 = x^3 - k has exactly 1 integral solution.
%C A179163 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
%H A179163 Jianing Song, <a href="/A179163/b179163.txt">Table of n, a(n) for n = 1..108</a> (using data from A179149)
%H A179163 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%F A179163 a(n) = A356713(n)^3. - _Jianing Song_, Aug 24 2022
%t A179163 (* 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 *)
%Y A179163 Cf. A081120, A081121, A179163-A179175, A356713.
%Y A179163 Complement of A179149 among the positive cubes.
%Y A179163 Cf. also A179145, A356703.
%K A179163 nonn
%O A179163 1,2
%A A179163 _Artur Jasinski_, Jun 30 2010
%E A179163 Edited and extended by _Ray Chandler_, Jul 11 2010