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A natural question about division

Source: 2019 Belarus Team Selection Test 5.1

September 2, 2019
number theoryfunctionalgebrapolynomial

Problem Statement

A function f:N→Nf:\mathbb N\to\mathbb N, where N\mathbb N is the set of positive integers, satisfies the following condition: for any positive integers mm and nn (m>nm>n) the number f(m)−f(n)f(m)-f(n) is divisible by m−nm-n. Is the function ff necessarily a polynomial? (In other words, is it true that for any such function there exists a polynomial p(x)p(x) with real coefficients such that f(n)=p(n)f(n)=p(n) for all positive integers nn?)
(Folklore)
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