On The Emergence of a New Prime Number And Omega Sequences

TL;DR

This paper explores the Omega sequence's role in number theory, its connection to a new prime, and its applications in primality testing and number representations, unifying various classical sequences.

Contribution

It introduces the Omega sequence as a unifying framework for prime numbers and classical sequences, with implications for primality tests and number theory.

Findings

Omega sequence linked to a new prime number

Unifies representations of classical number sequences

Applications to Lucas-Lehmer primality test and perfect numbers

Abstract

This paper highlights the emergence of the Omega sequence in number theory and its connection with the emergence of a new prime number, and also highlights its theoretical applications for Lucas-Lehmer primality test, and Euclid-Euler theory for even perfect numbers. We also show that Omega sequences unify and give new representations for Mersenne numbers, Fermat numbers, Lucas numbers, Fibonacci numbers, Chebyshev sequence, Dickson sequence, and others.