On product representations of squares

TL;DR

This paper investigates the size of subsets of integers avoiding k-element products that are perfect squares, providing new insights and answering a longstanding question about their asymptotic behavior for odd k ≥ 5.

Contribution

The paper uses probabilistic methods to show that for odd k ≥ 5, the largest such subset does not asymptotically occupy the entire set, answering Erdős's question negatively.

Findings

For odd k ≥ 5, the largest subset is significantly smaller than the entire set.

Probabilistic arguments demonstrate the non-trivial structure of these subsets.

Results contrast with previous conjectures about their asymptotic size.

Abstract

Fix $k \geq 2$. For any $N \geq 1$, let $F_k(N)$ denote the cardinality of the largest subset of $\{1,\dots,N\}$ that does not contain $k$ distinct elements whose product is a square. Erd\H{o}s, S\'ark\H{o}zy, and S\'os showed that $F_2(N) = (\frac{6}{\pi^2}+o(1)) N$, $F_3(N) = (1-o(1))N$, $F_k(N) \asymp N/\log N$ for even $k \geq 4$, and $F_k(N) \asymp N$ for odd $k \geq 5$. Erd\H{o}s then asked whether $F_k(N) = (1-o(1)) N$ for odd $k \geq 5$. Using a probabilistic argument, we answer this question in the negative.