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

Cody S. Hansen; Pace P. Nielsen

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

Number Theory 2022-10-26 v1

Authors: Cody S. Hansen , Pace P. Nielsen

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.

Keywords

cyclotomic polynomial prime factors odd perfect number parameterization Diophantine equation

Cite

@article{arxiv.2204.08971,
  title  = {Prime factors of $\Phi_3(x)$ of the same form},
  author = {Cody S. Hansen and Pace P. Nielsen},
  journal= {arXiv preprint arXiv:2204.08971},
  year   = {2022}
}

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