A103256 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^4.

2, 9, 18, 28, 32, 65, 70, 84, 105, 126, 144, 162, 211, 217, 260, 266, 273, 288, 344, 364, 386, 417, 448, 455, 456, 469

Offset

1,1

Comments

There are no solutions with (x,y,z) relatively prime. [Bruin]

If x is in this sequence, then so is x*m^4 for any m.

Links

Table of n, a(n) for n=1..26.

F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.

Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.

Wikipedia, Beal's conjecture.

Chai Wah Wu, all terms x in sequence such that x < 5*10^7 and there exists y with x^3+y^3 = z^4 <= 10^24.

Formula

A parametrized solution: (a (a^m+b^m))^m + (b(a^m+b^m))^m = (a^m+b^m)^(m+1) [From Wikipedia article - set m=3]. - James Mc Laughlin, Jan 28 2007

Example

x=9, y=18, 9^3 + 18^3 = 9^4, so 9 and 18 are on the list.
Other solutions are (2, 2, 2), (9, 18, 9), (28, 84, 28), (32, 32, 16), ...

Mathematica

xmax=500; p[z_] := p[z]=PowersRepresentations[z^4, 2, 3]; rep = {n1___, n2_ /; n2^4 <= xmax, n3___} :> Union[{n1, Sequence @@ Table[n2*k^4, {k, 1, Ceiling[(xmax/n2)^(1/4)]}], n3}]; sel = Union[ Flatten[ Select[ Table[p[z], {z, 1, 6 xmax/5}], Length[#] != 0 && 0 < #[[1, 1]] & ]]]; Take[ ReplaceRepeated[ sel, rep], 26] (* Jean-François Alcover, Sep 06 2011 *)

Prog

(Magma) [ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0, k^3])) | P[1] gt 0 and P[2] ne 0 and IsSquare(Abs(P[2]))} ]; // Geoff Bailey

Crossrefs

Sequence in context: A358946 A282519 A359580 * A028881 A294535 A294543

Adjacent sequences: A103253 A103254 A103255 * A103257 A103258 A103259

Keyword

nonn

Author

Cino Hilliard, Mar 20 2005

Extensions

Corrected and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using Magma, Jan 28 2007

a(9)-a(12) from Jonathan Vos Post, May 27 2007

Status

approved