2006/02/22 by Pace P. Nielsen, Pace Nielsen · 17 citations
Mathematics · #Advanced Mathematical Theories #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Perfect number #Perfect power #Prime (order theory) #Prime factor #Prime number #Prime power #math.NT #msc:11N25 #msc:11Y50
paper · pdf · doi:10.1090/s0025-5718-07-01990-4
published in Mathematics of Computation 76(260), 2109-2126 (American Mathematical Society) · 17 pages
arxiv created 2006/02/22 · openalex publication_date 2007/05/09 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
An odd perfect number, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is shown to have at least nine distinct prime factors. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 does-not-divide upper N"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo> ∤ </mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">3\nmid N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.