On a theorem of Kanold on odd perfect numbers

Abstract

We shall prove that ifis an odd perfect number such that p, q 1 , . . . , q r−1 are distinct primes, p ≡ α ≡ 1 (mod 4) and t divides 2β i + 1 for all i = 1, 2, . . . , r − 1, then t 5 divides N , improving an eighty-year old result of Kanold.