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%I #51 Jun 23 2023 23:28:24
%S 2,9,18,28,32,65,70,84,105,126,144,162,211,217,260,266,273,288,344,
%T 364,386,417,448,455,456,469
%N Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^4.
%C There are no solutions with (x,y,z) relatively prime. [Bruin]
%C If x is in this sequence, then so is x*m^4 for any m.
%H F. Beukers, <a href="http://dx.doi.org/10.1215/S0012-7094-98-09105-0">The Diophantine equation Ax^p+By^q=Cz^r</a>, Duke Math. J. 91 (1998), 61-88.
%H Nils Bruin, <a href="http://dx.doi.org/10.1007/10722028_9">On powers as sums of two cubes</a>, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Beal's_conjecture">Beal's conjecture</a>.
%H Chai Wah Wu, <a href="/A103256/a103256.txt">all terms x in sequence such that x < 5*10^7 and there exists y with x^3+y^3 = z^4 <= 10^24</a>.
%F 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
%e x=9, y=18, 9^3 + 18^3 = 9^4, so 9 and 18 are on the list.
%e Other solutions are (2, 2, 2), (9, 18, 9), (28, 84, 28), (32, 32, 16), ...
%t 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 *)
%o (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
%K nonn
%O 1,1
%A _Cino Hilliard_, Mar 20 2005
%E Corrected and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using Magma, Jan 28 2007
%E a(9)-a(12) from _Jonathan Vos Post_, May 27 2007
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