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%I #17 Aug 06 2022 07:23:08
%S 3,8,96,256,624,686,729
%N Positive integers x such that there exist positive integers y >= x and z satisfying x^3 + y^3 = z^5.
%C Warning! These terms have not been proved to be correct. There may be missing terms.
%C There are no solutions with (x,y,z) relatively prime. [Bruin]
%C For max(x,y) < 1.1*10^12, there are no more terms < 1458. Most likely this is true for all x,y. - _Chai Wah Wu_, Jan 15 2016
%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.
%e x=3, y=6, 3^3 + 6^3 = 3^5, so 3 is a term.
%e With max(x,y) < 10^4, we have these [x,y,z] triples: [3, 6, 3] [8, 8, 4] [96, 192, 24] [256, 256, 32] [729, 1458, 81] [1944, 1944, 108] [686, 2058, 98] [3696, 4368, 168] [3072, 6144, 192] [8192, 8192, 256] [2508, 8436, 228] ... - _David Broadhurst_
%e 624^3 + 14352^3 = 312^5. - _Chai Wah Wu_, Jan 11 2016
%Y See A103268 for another version.
%K more,nonn
%O 1,1
%A _N. J. A. Sloane_, Jan 31 2007
%E Term 624 added by _Chai Wah Wu_, Jan 11 2016