Prime factors of $\Phi_3(x)$ of the same form

TL;DR

This paper investigates the prime factorization structure of the cyclotomic polynomial 3(x) when it factors into primes, focusing on cases with 2, 3, or 4 prime factors to extend bounds related to odd perfect numbers.

Contribution

It provides a parameterization of solutions for the prime factorizations of 3(x) in specific cases, extending existing bounds on prime factors of odd perfect numbers.

Findings

Parameterization of solutions for 3(x) prime factorizations with 2, 3, or 4 factors

Extended bounds on the total number of prime factors of odd perfect numbers

Analysis of special cases simplifies understanding of 3(x) factorizations

Abstract

We parameterize solutions to the equality $\Phi_3(x)=\Phi_3(a_1)\Phi_3(a_2)\cdots\Phi_3(a_n)$ when each $\Phi_3(a_i)$ is prime. Our focus is on the special cases when $n=2,3,4$, as this analysis simplifies and extends bounds on the total number of prime factors of an odd perfect number.