p-adic Valuation of the Sum of Divisors

Abstract

For any positive integer n, let σ(n) be the sum of all positive divisors of n. For any prime p and any positive integer m, let ν _p(m) be the largest integer α such that p ^α ∣ m. Recently, Amdeberhan, Moll, Sharma and Villamizar proved that for any odd prime p and any integer n ≥ 2, ν _p(σ(n) ≤ [log _p n] if n satisfies some conditions, where [x] denotes the least integer not less than x. In this paper, for any odd prime p, we prove that ν _p(σ(n) ≤ [log _p n] for all positive integers n unconditionally. Moreover, we prove that if 3 ≤ p < 10⁵ is a prime and p ≠ 31, then there are only finitely many positive integers n such that ν _p(σ(n) = [log _p n]. For p = 31 and n ≥ 2, ν _p(σ(n) = [log _p n] if and only if n = 2⁴, 5², 2⁴5² and 2⁴5²(2 · 31 ^s − 1), where s is a positive integer and 2 · 31 ^s − 1 is a prime.

Keywords

Sum-of-divisors function / p-adic valuation / Nagell–Ljunggren equation / Goormaghtigh’s conjecture

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Peking University