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Functional Equation using number of divisors of n

Source: IMOC 2021 N8

August 11, 2021
algebrafunctional equation

Problem Statement

Find all integer-valued polynomials f,g:N→N such that  ∀  x∈N,τ(f(x))=g(x)f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x) holds for all positive integer xx, where τ(x)\tau (x) is the number of positive factors of xx
Proposed By - ckliao914
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