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Common prime divisor of a certain polynomial

Source: Korea junior mathematical olympiad 2nd round 2019 December

November 17, 2019
KJMOnumber theoryprime numbersalgebrapolynomial

Problem Statement

For prime number pp, prove that there are integers aa, bb, cc, dd such that for every integer nn, the expression n4+1āˆ’(n2+an+b)(n2+cn+d)n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right) is a multiple of pp.
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