Problem
NT-B2-M12-P003 Polynomial Residues Are Periodic
#3
★★☆☆☆ Level 2 of 5
Let \(f\in\mathbb Z[x]\), \(m\ge1\). Prove that the sequence \(f(0),f(1),f(2),\ldots\) modulo \(m\) has period \(m\).
Compare \(n+m\) and \(n\) modulo \(m\).
Since \(n+m\equiv n\pmod m\), for every monomial \((n+m)^k\equiv n^k\pmod m\). Hence \(f(n+m)\equiv f(n)\pmod m\). Thus the residues repeat with period \(m\), possibly not the minimal period.
The phrase “has period” is important; this need not be the minimal period.