Power savings for counting solutions to polynomial-factorial equations

Hung M. Bui; Kyle Pratt; Alexandru Zaharescu

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arXiv · 2204.08423

Power savings for counting solutions to polynomial-factorial equations

Hung M. Bui·Kyle Pratt·Alexandru Zaharescu

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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 $n\leq N$ 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 Pad\'e approximation.

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Hung M. Bui, Kyle Pratt, Alexandru Zaharescu. 2022-04-18. Power savings for counting solutions to polynomial-factorial equations. https://arxiv.org/abs/2204.08423

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