2025/12/02 by Joshua Cooper, Cooper, Joshua, Joseph Preuss +1
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2512.03188
We consider the Diophantine equation a!b! = c! due to Erdős, where we assume a ≤ b. It is widely believed that there are only finitely many nontrivial solutions, and considerable work has been dedicated to showing this. In one direction, Luca (2007) showed that the set of c's which can appear in solutions has density zero. Here we show that the set of a's appearing in solutions is also sparse. In particular, a cannot be one less than a large fraction of primes, and, under the assumption that √[k]a! \mod 1 is equidistributed in an appropriate sense, we show that the set of such a has asymptotic density zero.