Irrationality of the sum of a $p$-adic series

TL;DR

This paper proves that the sum of a specific p-adic series involving factorial valuations is irrational in the p-adic number system for all primes p.

Contribution

It establishes the irrationality of a particular p-adic series sum, extending understanding of p-adic series behavior.

Findings

The series sum is p-adic irrational for all primes p.

The proof applies p-adic valuation properties to demonstrate irrationality.

This work advances knowledge of p-adic series and their arithmetic properties.

Abstract

We prove that the sum of the series $\sum_{n=0}^{\infty}\, p^{v_p(n!)}$ is a $p$-adic irrational for all primes $p$, where $v_p(n!)$ denotes the exponent of the highest power of $p$ dividing $n!$.