# Odd perfect numbers have at least nine distinct prime factors

```@article{Nielsen2007OddPN,
title={Odd perfect numbers have at least nine distinct prime factors},
author={Pace P. Nielsen},
journal={Math. Comput.},
year={2007},
volume={76},
pages={2109-2126}
}```
An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 l N then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
Odd perfect numbers have a prime factor exceeding 108
• Mathematics
Math. Comput.
• 2008
Using the properties of cyclotomic polynomials, this work improves the result to show that every perfect number is divisible by a prime exceeding 108.
Perfect numbers - a lower bound for an odd perfect number
In this work we construct a lower bound for an odd perfect number in terms of the number of its distinct prime factors. We further generalize the formula for any natural number for which the number
On the Sum and Product of Distinct Prime Factors of an Odd Perfect Number
We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of
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.
BOUNDS FOR ODD k-PERFECT NUMBERS
• Mathematics
Bulletin of the Australian Mathematical Society
• 2011
Abstract Let k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is
Sieve methods for odd perfect numbers
• Mathematics
Math. Comput.
• 2012
It is proved that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10^8 < p < 10^1000.
On Odd Perfect Numbers and Even 3-Perfect Numbers
• Mathematics, Physics
Integers
• 2012
Abstract. An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there
On Dickson's Theorem Concerning Odd Perfect Numbers
It is shown that the number of even perfect numbers N is bounded by 4k2, which means that for each fixed natural number k, there are only finitely many odd perfectNumbers N with at most k distinct prime factors.
The Abundancy Index of Divisors of Odd Perfect Numbers
It is shown that n < q is sufficient for Sorli’s conjecture thatk = �q(N) = 1 to hold, and that q k < 2 n 2 , and that I(q k ) < I(n), where I(x) is the abundancy index of x.
Note on the Theory of Perfect Numbers
A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence

## References

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If N is an odd perfect number with k distinct prime factors then we show that N < 2 k . If some of the small prime factors of N are known then this bound can be further improved.
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An argument is outlined which demonstates that every odd perfect number which is not divisible by 3 has at least eleven distinct prime factors.
Outline of a proof that every odd perfect number has at least eight prime factors
An argument is outlined which demonstrates that every odd perfect number is divisible by at least eight distinct primes.
Odd perfect numbers have a prime factor exceeding 108
• Mathematics
Math. Comput.
• 2008
Using the properties of cyclotomic polynomials, this work improves the result to show that every perfect number is divisible by a prime exceeding 108.
The third largest prime divisor of an odd perfect number exceeds one hundred
It is proved that the third largest prime divisor of an odd perfect number must exceed 100.
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