On a theorem of Kanold on odd perfect numbers

Tomohiro Yamada ()
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Tomohiro Yamada: Osaka University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1278-1284

Abstract: Abstract We shall prove that if $$N=p^\alpha q_1^{2\beta _1} q_2^{2\beta _2} \cdots q_{r-1}^{2\beta _{r-1}}$$ N = p α q 1 2 β 1 q 2 2 β 2 ⋯ q r - 1 2 β r - 1 is an odd perfect number such that $$p, q_1, \ldots , q_{r-1}$$ p , q 1 , … , q r - 1 are distinct primes, $$p\equiv \alpha \equiv 1\ \left( \textrm{mod}\ 4\right) $$ p ≡ α ≡ 1 mod 4 and t divides $$2\beta _i+1$$ 2 β i + 1 for all $$i=1, 2, \ldots , r-1$$ i = 1 , 2 , … , r - 1 , then $$t^5$$ t 5 divides N, improving an eighty-year old result of Kanold.

Keywords: Odd perfect number; Arithmetic function; Kanold’s theorem; Exponential diophantine equation; Primary 11A25; Secondary 11A05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00530-y

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