# Upper bounds on the second largest prime factor of an odd perfect number

@article{Zelinsky2019UpperBO,
title={Upper bounds on the second largest prime factor of an odd perfect number},
author={Joshua Zelinsky},
journal={International Journal of Number Theory},
year={2019}
}
• Joshua Zelinsky
• Published 28 October 2018
• Mathematics
• International Journal of Number Theory
Acquaah and Konyagin showed that if [Formula: see text] is an odd perfect number with prime factorization [Formula: see text], where [Formula: see text], then one must have [Formula: see text]. Using methods similar to theirs, we show that [Formula: see text] and that [Formula: see text] We also show that if [Formula: see text] and [Formula: see text] are close to each other, then these bounds can be further strengthened.
3 Citations
On the small prime factors of a non-deficient number
Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd
On the third largest prime divisor of an odd perfect number
• Mathematics
• 2019
Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a
ON ODD PERFECT NUMBERS
In this note, we introduce the notion of the disc induced by an arithmetic function and apply this notion to the odd perfect number problem. We show that under certain special local condition an odd

## References

SHOWING 1-8 OF 8 REFERENCES
The second largest prime divisor of an odd perfect number exceeds ten thousand
The latter bound of the statement in the title of this paper is improved, showing that the largest prime divisor of an odd perfect number must exceed 10 6 , and Hagis showed that the second largest must exceeds 10 3 .
On the number of prime factors of an odd perfect number
• Mathematics
Math. Comput.
• 2014
It is shown that an odd perfect number N is of the form N = pᶱm² where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m means p ≥ 2ω(N) − 1.
A note on odd perfect numbers
• Mathematics
ArXiv
• 2011
It is shown that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{alpha})/q^(\alpha}>5$ is bounded by some function depending on $K$.
Solving the Odd Perfect Number Problem: Some Old and New Approaches
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)
An Improvement of an Inequality of Ochem and Rao Concerning Odd Perfect Numbers
It is shown that if $3 |N$ is not| N, then $\Omega(N) \geq \frac{8}{3}\omega(N)-\frac{7}{3}$ and if “3 | N” is not |N, then “Omega (N)” = 21, where “N” denotes the number of distinct prime divisors of n.
On the radical of a perfect number
• Mathematics
• 2010
In this note, we look at the radical (that is, the squarefree kernel) of perfect numbers. We raise the question of whether large per- fect numbers have the tendency to become far apart from each
On Dris Conjecture about Odd Perfect Numbers
The Euler's form of odd perfect numbers, if any, is $n=\pi^{\alpha}N^2$, where $\pi$ is prime, $(\pi,N)=1$ and $\pi\equiv \alpha \equiv 1 \pmod{4}$. Dris conjecture states that $N>\pi^{\alpha}$. We
ON PRIME FACTORS OF ODD PERFECT NUMBERS
• Mathematics
• 2012
We prove that a prime factor q of an odd perfect number x satisfies the inequality q < (3x)1/3.