

A103256


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


1



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
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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*a^4 for any a.


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), 6188.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169184, 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 McLaughlin, 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] (* JeanFrançois 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: A083707 A240651 A282519 * 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



