Solving the Odd Perfect Number Problem: Some Old and New Approaches

Abstract

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = \sigma(p^k)/p^k, Y = \sigma(m^2)/m^2$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.85 < X + Y < 3$, is disproved. Various results on the abundancy index and solitary numbers are used in the disproof. Numerical evidence against the said conjecture will likewise be discussed. We will show that if an OPN $N$ has the form above, then $p^k < (2/3){m^2}$ follows from \cite{D10}. We will also attempt to prove a conjectured improvement of this last result to $p^k < m$ by observing that $\sigma(p^k)/m \neq 1$ and $\sigma(p^k)/m \neq \sigma(m)/p^k$ in all cases. Lastly, we also prove the following generalization: If $N = \displaystyle\prod_{i=1}^r {{p_i}^{{\alpha}_i}}$ is the canonical factorization of an OPN $N$, then $$\sigma({p_i}^{{\alpha}_i}) \leq (2/3){\frac{N}{{p_i}^{{\alpha}_i}}}$$ for all $i$. This gives rise to the inequality $$N^{2 - r} \leq (1/3)(2/3)^{r - 1}$$ which is true for all $r$, where $r = \omega(N)$ is the number of distinct prime factors of $N$.