Search: id:A103256
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%I A103256
%S A103256 2,9,18,28,32,65,70,84,105,126,144,162
%N A103256 Positive integers x such that there exist positive integers y and z satisfying 
               x^3 + y^3 = z^4.
%C A103256 There are no solutions with (x,y,z) relatively prime. [Bruin]
%D A103256 F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 
               (1998), 61-88.
%D A103256 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.
%H A103256 Wikipedia, <a href="http://en.wikipedia.org/wiki/Beal's_conjecture">Beal's 
               conjecture</a>
%F A103256 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
%e A103256 x=9, y=18, 9^3 + 18^3 = 9^4, so 9 and 18 are on the list.
%e A103256 Other solutions are (2, 2, 2), (9, 18, 9), (28, 84, 28), (32, 32, 16), 
               ...
%o A103256 (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]))} ]; (from 
               Geoff Bailey)
%Y A103256 Sequence in context: A083419 A126082 A083707 this_sequence A028881 A083708 
               A037421
%Y A103256 Adjacent sequences: A103253 A103254 A103255 this_sequence A103257 A103258 
               A103259
%K A103256 nonn
%O A103256 1,1
%A A103256 Cino Hilliard (hillcino368(AT)gmail.com), Mar 20 2005
%E A103256 Corrected and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using 
               MAGMA, Jan 28 2007.
%E A103256 a(9)-a(12) from Jonathan Vos Post (jvospost3(AT)gmail.com), May 27 2007

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