# 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} }

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

- Mathematics
- 2020

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

- Mathematics
- 2021

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

- MathematicsMath. Comput.
- 1999

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

- MathematicsMath. 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

- MathematicsArXiv
- 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

- Mathematics
- 2012

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

- MathematicsIntegers
- 2018

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

- Mathematics
- 2017

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.