Odd perfect numbers have at least nine distinct prime factors

DOI10.1090/S0025-5718-07-01990-4zbMath1142.11086arXivmath/0602485WikidataQ56059210 ScholiaQ56059210MaRDI QIDQ3592696

Pace P. Nielsen

Publication date: 13 September 2007

Published in: Mathematics of Computation (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0602485

zbMATH Keywords

branch-and-cut algorithmsum of divisorsarithmetical functionodd perfect numbers

Mathematics Subject Classification ID

Arithmetic functions; related numbers; inversion formulas (11A25) Values of arithmetic functions; tables (11Y70)

Related Items (19)

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Cites Work

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• On the number of distinct prime factors of an odd perfect number
• Multiply perfect numbers, Mersenne primes, and effective computability
• The non-existence of odd perfect numbers of a certain form
• Über ungerade vollkommene Zahlen
• New Solutions of a p - 1 ≡1 (mod p 2 )
• Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors
• On Odd Perfect, Quasiperfect, and Odd Almost Perfect Numbers
• Improved Techniques for Lower Bounds for Odd Perfect Numbers
• The second largest prime divisor of an odd perfect number exceeds ten thousand
• Odd perfect numbers
• Solutions of the congruence 𝑎^{𝑝-1}≡1 (mod 𝑝^{𝑟})
• More on the total number of prime factors of an odd perfect number
• Odd Perfect Numbers Not Divisible by 3. II
• Sketch of a Proof that an Odd Perfect Number Relatively Prime to 3 has at Least Eleven Prime Factors
• Odd perfect numbers have a prime factor exceeding $10^{7}$
• The third largest prime divisor of an odd perfect number exceeds one hundred
• Odd perfect numbers are divisible by at least seven distinct primes

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