On the 2-adic valuation of (a^b-c^d)

TL;DR

This paper establishes an inequality for the 2-adic valuation of expressions like a^b - c^d, enabling divisibility analysis without direct computation, with implications for number theory and computer science.

Contribution

It introduces a new inequality for v_2(a^b - c^d), facilitating divisibility analysis and exploring 2-adic behavior with applications in informatics.

Findings

Derived an inequality for v_2(a^b - c^d)

Analyzed the asymptotic behavior of the function

Linked 2-adic valuation to binary systems and applications

Abstract

In this paper will be proved an inequality regarding $v_2(a^{b}-c^{d})$. Using this formula it will be possible to have informations about the divisibility of 2 of this function without computing it. Then, will be studied the behavior of this function when it approaches infinity and there will be a lot of analogies with 2-adic integers. This result may be easily used on a calculator, to facilitate the analysis of the divisibility by 2 of this function for large values. Furthermore 2-adic valuation and in general 2-adic analysis is related to binary system, and every work related to it could have a lot of applications in informatics.