On the finiteness of solutions for polynomial-factorial Diophantine equations

Wataru Takeda

Published 2019-03-04Version 1

We study the number of integer solutions $(x,y,l)$ of an equation $F(x,y)=\Pi_K(l)$, where $F(x,y)$ is a homogeneous polynomial with integer coefficients and $\Pi_K(l)$ is a generalized factorial function over number fields. We show a necessary condition for the existence of infinitely many solutions. As a corollary, we obtain the finiteness of solution for $P(x)=l!$, where $P$ is decomposed into irreducible polynomials of even degree.