Solving the Odd Perfect Number Problem: Some New Approaches

Abstract

A conjecture predicting an injective and surjective mapping $X=\frac{\sigma (p^k)}{p^k},Y=\frac{\sigma (m^2)}{m^2}$ 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. We will show that if an OPN $N$ has the form above, then $p^k<2/3m^2$. We then give a somewhat weaker corollary to this last result ($m^2-p^k\ge 8$) and give possible improvements along these lines. We will also attempt to prove a conjectured improvement to $p^k<m$ by observing that $\frac{\sigma (p^k)}{m}\ne 1$ and $\frac{\sigma (p^k)}{m}\ne \frac{\sigma (m)}{p^k}$ in all cases. Lastly, we also prove the following generalization: If $N=\prod _{i=1}^r{p_i}^{{\alpha }_i}$ is the canonical factorization of an OPN $N$, then $\sigma ({p_i}^{{\alpha }_i})\le (2/3)\frac{N}{{p_i}^{{\alpha }_i}}$ for all $i$. This gives rise to the inequality $N^{2-r}\le (1/3){(2/3)}^{r-1}$, which is true for all $r$, where $r=\omega (N)$ is the number of distinct prime factors of $N$.

arXiv comment3 pages, Electronic Proceedings of the 11th Science and Technology Congress, De La Salle University, Manila, Philippines, Sept. 22, 2009

References in corpus (1)

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