Diophantine problems related to the Omega constant

TL;DR

This paper explores diophantine problems associated with the Omega constant, Lambert W-function values, and their p-adic extensions, aiming to understand their number-theoretic properties and relationships.

Contribution

It introduces new diophantine problems related to special constants and functions, extending the scope to p-adic contexts and providing initial insights into their properties.

Findings

Formulated diophantine problems for Omega constant and Lambert W-values

Extended analysis to p-adic versions of these constants

Provided preliminary results on their number-theoretic behavior

Abstract

Some diophantine problems are stated for the Omega constant and, more generally, the values of Lambert $W$-function and their $p$-adic extensions.