Submit progress

on Furstenberg-Sárközy Problem (AlphaEvolve problem 31)

The statement

If $k, m \geq 2$ and $N \geq 1$ , let $C(k,N)$ (resp. $C(k,\Z/M\Z)$ ) denote the size of the largest subset of ${1,\dots,N}$ that does not contain any two elements that differ by a perfect $k^{\mathrm{th}}$ power. Establish upper and lower bounds for $C(k,N)$ and $C(k,\Z/M\Z)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.