Sieve methods for odd perfect numbers

Abstract

Abstract. Using a new factor chain argument, we show that 5 does not divide an odd perfect number indivisible by a sixth power. Applying sieve techniques, we also find an upper bound on the smallest prime divisor. Putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10 8 < p < 10 1000 . These results are generalized to much broader situations.

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

“…To prove Inequality 46 we then add our inequalities as follows: 4 ||N then one must have at least one prime which is not the special prime raised to at least the 6th power, and hence have u ≥ 1. We thus may adjust the above equations slightly: If (N, 55) = 1, we have instead of 47, we have ω = s + t + u − 1 and f 5 = f 11 = 0.…”

A Class of New Near-Perfect Numbers

“…For odd perfect numbers, Euler obtained a necessary condition for the existence (see [5]). In recent years, there have been many papers for odd perfect numbers having to do with the conjecture that there exists no odd perfect numbers (see [2,3,4,7,12]). Until now, the conjecture has not been proved.…”