Power savings for counting solutions to polynomial-factorial equations

2022/04/18 by Hung M. Bui, Kyle Pratt, Bui, Hung M. +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2204.08423

openalex publication_date 2022/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions n≤ N to n! = P(x) which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is o(N). The proof uses techniques of Diophantine and Padé approximation.

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