5

Part of 2019 Korea Junior Math Olympiad.

Problems(1)

Common prime divisor of a certain polynomial

Source: Korea junior mathematical olympiad 2nd round 2019 December

For prime number

p

, prove that there are integers

a

b

c

d

such that for every integer

n

, the expression

n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)

is a multiple of

p

KJMOnumber theoryprime numbersalgebrapolynomial