More on the Nonexistence of Odd Perfect Numbers of a Certain Form

2015/12/03 by Patrick Brown, P. W. F. Brown, Brown, Patrick
Mathematics · #11-04 #11A25 #11B83 #11Y55 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11-04 #msc:11A25 #msc:11B83 #msc:11Y55

paper · pdf · doi:10.48550/arxiv.1512.01270

9 pages

arxiv created 2015/12/03 · openalex publication_date 2015/12/03 · arxiv updated 2015/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Euler showed that if an odd perfect number exists, it must be of the form N = p^αq₁^2β₁ … q_k^2β_k, where p, q₁, …, q_k are distinct odd primes, α, β_i ≥ 1, for 1 ≤ i ≤ k, with p ≡ α≡ 1 \pmod4. In 2005, Evans and Pearlman showed that N is not perfect, if 3|N or 7|N and each β_i ≡ 2 \pmod5. We improve on this result by removing the hypothesis that 3|N or 7|N and show that N is not perfect, simply, if each β_i ≡ 2 \pmod5.

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