AlphaEvolve problems·alphaevolve:31
Furstenberg-Sárközy Problem
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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.
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