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

1,1

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

Table of n, a(n) for n=1..9.

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

See A114737 for another version.

Sequence in context: A137129 A098106 A059361 * A114157 A021275 A281802

Adjacent sequences: A103265 A103266 A103267 * A103269 A103270 A103271

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