On Odd Perfect Numbers and Even 3-Perfect Numbers

Abstract

An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there are no even 3-perfect numbers of the form 2 a M , where M is odd and squarefree and a Ä 718, besides the six known examples.

Connected components of complex divisor functions

“…Using the fact that c (t) = t c + O(t 2c ), one can verify that arg( c (uτ )/ c (τ )) → ψ 0 /3 as τ → ∞. Consequently, there exists τ 0 ≥ 0 such that (3) arg( c (uτ )/ c (τ )) < ψ 0 /2 for all τ ≥ τ 0 .…”

“…Many early theorems in number theory spawned from attempts to understand perfect numbers. Although few modern mathematicians continue to attribute the same theological or mystical significance to perfect numbers that ancient people once did, these numbers remain a substantial inspiration for research in elementary number theory [2,3,5,11,[14][15][16]18]. Around 100 A.D., Nicomachus stated that perfect numbers represent a type of "harmony" between "deficient" and "abundant" numbers.…”

On the quantity I(q^k) + I(n^2) where q^k n^2 is an odd perfect number

“…• Case 2. If q = 5 and k > 1, then using the facts that q k = 5 5 (see the paper by Cohen and Sorli [1]), and k ≡ 1 (mod 4), we obtain 5 9 | q k , so that I(5 9 ) ≤ I(q k ) < 5 4 < 8 5 < I(n 2 ) ≤ 2 I(5 9 )…”

Some modular considerations regarding odd perfect numbers – Part II

“…where we have used the inequality σ((p + 1)/2) ≤ p − 1 (from the line immediately preceding the statement of Theorem 4 in page 5 of Cohen and Sorli's paper [4]). This results in the trivial lower bound p ≥ 5 -hence, still no contradiction, at this point.…”