Odd perfect numbers have a prime factor exceeding $10^8$

Abstract

Abstract. Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 10 7 . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding 10 8 .

On the number of total prime factors of an odd perfect number

“…This proposition stops at 397 is because the relevant product is actually greater than 2 for the next prime, 401. The result could be extended if the result from [6] could be extended further; however, extending that result (say to that an odd perfect number must have a prime divisor which is at least 10 9 ) would likely take either very heavy new computations or would take some fundamental new insight. That said, it is plausible that a similar result could be proved just for odd perfect numbers not divisible by any prime less than some bound, and this would allow one to extend the above proposition in this case.…”

“…Combining Equation 113 with Nielsen's upper bound 114, as well as the fact that the largest prime factor of an odd perfect number must be at least 10 8 by [6] we get that (5 which is is a linear inequality restricting f 5 and f 11 but it is much too weak to give a useful restriction for improving the constant. There appear to be four possible approaches to improving this inequality.…”

“…Alternatively, one must have 5 6 |N or 11 6 |N if either 5|N or 11|N. We can without too much work check that all of these force a bound at least as tight as 77.…”

Measuring Abundance with Abundancy Index

“…The first such lower bound was proved by Haggis [25], who proved every Odd Perfect Number has a Prime Factor which exceeds 10 6 . Iannucci [26] [27], Jenkins [28], Goto and Ohno [29] proved that the largest three factors must be at least 100000007, 10007, and 101 [12].…”

Improved Upper Bounds for Odd Multiperfect numbers

“…Recently, Ochem and Rao [9] proved that N is greater than 10 1500 . In 2008, Goto and Ohno [4] proved that N has a prime factor exceeding 10 8 . In 1913, Dickson [2] proved that there are only finitely many odd perfect numbers with r distinct prime factors.…”