OFFSET
1,1
COMMENTS
Warning! These terms have not been proved to be correct. There may be missing terms. - N. J. A. Sloane, Jan 05 2007
There are no solutions with (x,y,z) relatively prime. [Bruin]
Trivially, if m^3 + n^3 = z^2, then (z*m)^3 + (z*n)^3 = z^5. So from A103254 we can find many solutions. - James Mc Laughlin, Jan 30 2007
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
LINKS
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.
EXAMPLE
1 + 2^3 = 3^2 so 3^3 + 6^3 = 3^5 and 3 and 6 are terms.
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]
624^3 + 14352^3 = 312^5. - Chai Wah Wu, Jan 11 2016
MATHEMATICA
r[z_] := Reduce[x > 0 && y > 0 && x^3 + y^3 == z^5, {x, y}, Integers];
sols = Reap[Do[rz = r[z]; If[rz =!= False, xyz = {x, y, z} /. {ToRules[rz]}; Print[xyz]; Sow[xyz]], {z, 1, 1000}]][[2, 1]] // Flatten[#, 1]&;
sols[[All, 1]] // Union (* Jean-François Alcover, Oct 18 2019 *)
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Cino Hilliard, Mar 20 2005
EXTENSIONS
Corrected by David Broadhurst and others, Jan 30 2007
Term 624 added by Chai Wah Wu, Jan 11 2016
STATUS
approved