Outline of a Proof that Every Odd Perfect Number has at Least Eight Prime Factors

Abstract

Abstract.An argument is outlined which demonstrates that every odd perfect number is divisible by at least eight distinct primes.

Maple explorations, perfect numbers, and Mersenne primes

“…Brent, Cohen and Te Riele [4] have shown that there is no odd perfect number below 10^*"*. Moreover, Hagis [5] has shown (his announcement of this result [6] came a few years earlier in 1975) that an odd perfect number must have at least 8 prime factors. Furthermore, Heath-Brown [7] has shown that if n is an odd perfect number with at most k prime factors, then n < 4'\…”

“…> prevprime(\OO); 97 We can let Maple list for us the first 50 prime numbers: > s:=NULL: for i to 50 do s:=s,ithprime(i)od:s; 2, 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229 Generating Bernoulli numbers:…”

On some properties of the Euler's factor of certain odd perfect numbers

“…We note that the number of distinct prime factors (n) of an odd perfect number n is (n) 8 [2] (if 3 does not divide n, then (n) 11, [3,4]). A recent result of this kind is: if (n) = 8, then 5|n [8].…”

Odd perfect numbers

“…We now take & to be the smallest integer i for which (3) holds. Thus (4) and so, in the case k = r, we have…”

“…However it is known that there is no such number below 10 300 (see Brent[l]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] who showed that if JV is an odd perfect number with at most k prime factors, then JV < (4&) (4 * )!!…”