What are the two smallest positive whole numbers the difference of whose squares is a cube and the difference of whose cubes is a square ?

(In reply to And after that there's... by Charlie)

"After 6 and 10 come 384 and 640." Here 6 and 10 are smaller than 384 and 640. But the problem asks for "the two smallest positive whole numbers" and assuming nontriviality of the solution, how do we know that such actually exist? For example, could there be a y such that 2 and y are a nontrivial solution? If 6 and 10 are indeed the two smallest, then we need to somehow show that every other solution x,y with y > x has x > 6 and y > 10. I don't think this can be done with a computer program.

Posted by Richard on 2003-12-06 21:44:54