A103268 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^5.
3, 6, 8, 96, 192, 256, 624, 686, 729
Offset: 1
Examples
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
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.
Crossrefs
See A114737 for another version.
Programs
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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 *)
Extensions
Corrected by David Broadhurst and others, Jan 30 2007
Term 624 added by Chai Wah Wu, Jan 11 2016
Comments