# 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.

## 50 Citations

Odd perfect numbers have a prime factor exceeding 108

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Using the properties of cyclotomic polynomials, this work improves the result to show that every perfect number is divisible by a prime exceeding 108.

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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

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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

- MathematicsBulletin of the Australian Mathematical Society
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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.

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On Dickson's Theorem Concerning Odd Perfect Numbers

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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

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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

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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…

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