Abstract
This thesis relates some results about odd perfect numbers and factorizations of cyclotomic polynomials evaluated on integers. In Chapter 2 we parameterize solutions to the equality $\Phi_3(x)=\Phi_3(a_1)\Phi_3(a_2)\cdots\Phi_3(a_n)$, where $x,a_i\in \Z_{>0}$ for $1\leq i\leq n$, $\Phi_3$ is the third cyclotomic polynomial, and each $\Phi_3(a_i)$ is prime, in the cases that $n=2,3,4,5$. Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number (counting multiplicity) of prime factors of $N$. In Chapter 3, we use a result from Chapter 2 to show that $\frac{99}{37}\omega(N) - \frac{187}{37} \leq \Omega(N)$ and that if $3\nmid N$, then $\frac{51}{19}\omega(N)-\frac{46}{19} \leq \Omega(N)$.
Degree
MS
College and Department
Computational, Mathematical, and Physical Sciences; Mathematics
Rights
https://lib.byu.edu/about/copyright/
BYU ScholarsArchive Citation
Hansen, Cody Stenberg, "Prime Factors of the Third Cyclotomic Polynomial of the Same Form and Inequalities Related to Odd Perfect Numbers" (2025). Theses and Dissertations. 10748.
https://scholarsarchive.byu.edu/etd/10748
Date Submitted
2025-04-21
Document Type
Thesis
Handle
http://hdl.lib.byu.edu/1877/etd13584
Keywords
Odd perfect numbers, Diophantine equations
Language
english