On the value set of \(n!m!\) modulo a large prime (Q1012404)

Questions involving the distribution of values of \(n!\) modulo \(p\) for large primes \(p\) have generated some interest lately. For example, it is not known that the sequence of factorials occupies a positive proportion modulo \(p\) as \(p\) becomes large. However, \textit{M. Z. Garaev, F. Luca} and \textit{I. E. Shparlinski} [Trans. Am. Math. Soc. 356, No. 12, 5089--5102 (2004; Zbl 1060.11046)] showed that if one looks at the residue classes of the form \(m!n!\) for some positive integers \(m\) and \(n\) modulo \(p\), then these occupy at least \((5/8+o(1))p\) distinct residue classes modulo \(p\) as \(p\) tends to infinity. The proportion \(5/8\) was improved by \textit{Y.-G. Chen} and \textit{L.-X. Dai} to \(3/4\) in [Integers 6, Paper A21, 3 p., electronic only (2006; Zbl 1114.11020)]. In this short note, the author presents the further improvement of \(41/48\). The method of proof combines elementary arguments with deep results from the theory of character sums like Weil's estimate for hybrid character sums with rational arguments.