A114737 Positive integers x such that there exist positive integers y >= x and z satisfying x^3 + y^3 = z^5.

3, 8, 96, 256, 624, 686, 729

Offset

1,1

Comments

Warning! These terms have not been proved to be correct. There may be missing terms.

There are no solutions with (x,y,z) relatively prime. [Bruin]

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

Table of n, a(n) for n=1..7.

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

x=3, y=6, 3^3 + 6^3 = 3^5, so 3 is a term.
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
624^3 + 14352^3 = 312^5. - Chai Wah Wu, Jan 11 2016

Crossrefs

See A103268 for another version.

Sequence in context: A136309 A266671 A069703 * A233163 A281709 A099296

Adjacent sequences: A114734 A114735 A114736 * A114738 A114739 A114740

Keyword

more,nonn

Author

N. J. A. Sloane, Jan 31 2007

Extensions

Term 624 added by Chai Wah Wu, Jan 11 2016

Status

approved