Difference of powers of consecutive primes which are perfect squares

TL;DR

This paper explores the properties of consecutive primes where the difference of their powers results in perfect squares, proposing conjectures about these special prime pairs.

Contribution

It introduces new conjectures regarding the existence of natural number pairs for which the difference of prime powers is a perfect square.

Findings

Identifies specific prime pairs with perfect square differences of powers

Proposes conjectures for the general behavior of such prime pairs

Provides initial evidence and theoretical insights into prime power differences

Abstract

We investigate the consecutive primes $p$ and $q$ ($p > q$) for which there exists a pair of natural numbers $(x,y)$ such that $p^x-q^y$ is a perfect square and make some conjectures.