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.
Original language | English (US) |
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Article number | 109021 |
Journal | Advances in Mathematics |
Volume | 422 |
DOIs | |
State | Published - Jun 1 2023 |
Keywords
- Padé approximation
- Polynomial-factorial equation
- Simultaneous rational approximation
ASJC Scopus subject areas
- General Mathematics