Square products of factorials and a conjecture of Erd\H{o}s and Graham

Abstract

For $n\ge2$ let $F(n)$ be the least $k\ge2$ such that $n!$ is the largest factor in a product of $k$ distinct factorials that is a perfect square, and let $D_k(X)$ be the number of $n\le X$ with $F(n)=k$. Erdos and Graham asked for the order of growth of $D_k(X)$ for $3\le k\le6$, and conjectured that $D_6(X)\gg X$. We prove that $D_3(X)=\kappa_3\sqrt X+O_\varepsilon(X^{2/5+\varepsilon})$ with an explicit constant $\kappa_3=2.7097\ldots$, and that $D_5(X)\asymp D_6(X)\asymp X$. Together with classical facts, this determines the order of growth of $D_k(X)$ for every $k$. The exponent $2/5$ comes from balancing a uniform bound for Pell equations against Gallagher's larger sieve, with residue restrictions supplied by the Weil bound. For five and six factors we restrict to integers with a prime factor exceeding $X^{1-\alpha}$, where $\alpha>0$ is small and fixed. We exclude shorter representations by combining an equidistribution estimate for primes of Matomaki, Radziwill, Shao, Tao and Teravainen with the large sieve. In an appendix we use a zero-sum theorem for finite abelian groups to construct, for every $m\ge2$, perfect $m$-th powers that are products of a bounded number of distinct factorials with arguments given by fixed affine functions.