Odd perfect numbers have a prime factor exceeding 10^9

Abstract

We prove that every odd perfect number is divisible by a prime exceeding 10^9, improving the bound 10^8 of Goto and Ohno. The proof follows the method of Hagis and Cohen, Jenkins, and Goto and Ohno. Its computational part is organised so that every intermediate claim is either an exact inequality checked in integer arithmetic or a finite list that can be regenerated independently; the implementation reproduces all previously published bounds before being applied at 10^9.