MathDB
Polynomial and divisors

Source: 2016 Taiwan TST Round 3

July 26, 2016
algebrapolynomialnumber theoryprime divisor

Problem Statement

Let f(x)f(x) be the polynomial with integer coefficients (f(x)f(x) is not constant) such that (x3+4x2+4x+3)f(x)=(x3−2x2+2x−1)f(x+1)(x^3+4x^2+4x+3)f(x)=(x^3-2x^2+2x-1)f(x+1) Prove that for each positive integer n≥8n\geq8, f(n)f(n) has at least five distinct prime divisors.
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