Odd perfect numbers, Diophantine equations, and upper bounds

Abstract

We obtain a new upper bound for odd multiperfect numbers. If N is an odd perfect number with k distinct prime divisors and P is its largest prime divisor, we find as a corollary that 10 12 P 2 N < 2 4 k . Using this new bound, and extensive computations, we derive the inequality k ≥ 10.

The abundancy index of divisors of odd perfect numbers – Part II

“…Ochem and Rao [9] recently proved that N > 10 1500 . Nielsen [8] obtained the lower bound ω(N ) ≥ 10 for the number of distinct prime factors of N , improving on his last result ω(N ) ≥ 9 (see [7]).…”

A new upper bound for numbers with the Lehmer property and its application to repunit numbers

“…The key ingredient in the proof of the upper bound for numbers with the Lehmer property is the following lemma given by Nielsen in his proof of the bound for odd perfect numbers. Lemma 2.1 (Nielsen, [7]). Let r, a, b ∈ N and x 1 , .…”

“…Pomerance showed in [8] that if n satisfies the Lehmer property, then n is squarefree and n < K 2 K , where K denotes the number of prime divisors of n. Moreover Renze in [9] gave a bound K ≥ 15. If additionally 3 | n, then K ≥ 40 · 10 6 and n > 10 36·10 7 .…”

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

“…Nielsen [12] has proved that if N is an odd perfect number with k distinct prime factors and largest prime divisor P then…”

“…Note that Ochem and Rao's second inequality is stronger than the first as long as ω(N) ≤ 81. Nielsen [12] has shown that ω(n) ≥ 10. In a previous paper [23], the author improved on Ochem and Rao's result, showing that:…”