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NT Functional with factorial and exponent

Source: Mexican Quarantine Mathematical Olympiad P5

April 26, 2020
factorialnumber theoryfunctional equation

Problem Statement

Let N={1,2,3,… }\mathbb{N} = \{1, 2, 3, \dots \} be the set of positive integers. Find all functions f:N→Nf:\mathbb{N}\rightarrow \mathbb{N}, such that for all positive integers nn and prime numbers pp: p∣f(n)f(p−1)!+nf(p).p \mid f(n)f(p-1)!+n^{f(p)}.
Proposed by Dorlir Ahmeti
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