Prime factors of ${\Phi }_3(x)$ of the same form

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.