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

TL;DR

This paper explores old and new approaches to the long-standing open problem of whether odd perfect numbers exist, including disproving a specific conjecture and establishing new inequalities related to their prime factorization.

Contribution

It introduces novel inequalities and disproves a conjecture linking the abundancy index to rational points on a hyperbola, advancing understanding of the structure of odd perfect numbers.

Findings

Disproved a conjecture relating OPNs to rational points on a hyperbolic arc.

Established inequalities involving prime power factors of OPNs.

Provided bounds on the sum of divisors and the number of prime factors of OPNs.

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 $N$.