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COMMENTS
| Warning! These terms have not been proved to be correct. There may be missing terms. - N. J. A. Sloane (njas(AT)research.att.com), Jan 05 2007
There are no solutions with (x,y,z) relatively prime. [Bruin]
Trivially, if m^3+n^3 = z^2, then (zm)^3+(zn)^3 = z^5. So from A103254 we can find many solutions. - James McLaughlin, Jan 30 2007
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REFERENCES
| 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.
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EXAMPLE
| 1+2^3=3^2 so 3^3+6^3=3^5 and 3 and 6 are members.
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, Jan 30 2007
These are variously immediate consequences of 1+1=2, 1+2^3=3^2, 1+3^3=2^2*7 and, much more unexpectedly, 11^3+37^3=2^4*3^2*19^2. The last example shows that solutions with a common factor are not completely trivial. [Comment based on email from Alf van der Poorten, Feb 15 2007]
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