Let k be a positive integer.
Prove that there must exist a positive integer n such that the sets A( A={1^2, 2^2, 3^2, . . . })
and B ( B = {1^2+n, 2^2+n, 3^2+n, . . . }) share k identical members.
The task is to pick n so that a^2 = b^2 + n has k solutions for a given k when a and b range over the positive integers.
Looking at (a + b)(a  b) = n, all that is needed is to pick n big enough.
Powers of k suggest themselves. n=k^k isn't big enough and n=k^2k fails for even k, but n=k^3k allows at least k matches.

Posted by xdog
on 20181231 08:48:09 