Finiteness of trivial solutions of factorial products yielding a factorial over number fields

TL;DR

This paper investigates the finiteness of trivial solutions to factorial-related Diophantine equations over number fields, employing a Bertrand type estimate for primes that split completely, and excludes the rational case.

Contribution

It establishes the finiteness of trivial solutions over number fields using a new application of prime splitting estimates, extending previous results beyond the rational case.

Findings

Finiteness of trivial solutions over number fields proven

Application of Bertrand type estimate to factorial equations

Excludes the rational number field case

Abstract

We consider a Bertrand type estimate for primes splitting completely. As one of its applications, we show the finiteness of trivial solutions of Diophantine equation about the factorial function over number fields except for the case the rational number field.