Some new results on odd perfect numbers

G. G. Dandapat, John L. Hunsucker and Carl Pomerance

Vol. 57 (1975), No. 2, 359–364

DOI: 10.2140/pjm.1975.57.359

Abstract

If m is a multiply perfect number (σ(m) = tm for some integer t), we ask if there is a prime p with m = p^an, (p^a,n) = 1,σ(n) = p^a, and σ(p^a) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers necessarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) = 2n.

Mathematical Subject Classification

Primary: 10A40

Milestones

Received: 16 October 1974

Published: 1 April 1975