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integer f, f(n),f(n+1),... , f(n+k-1) are not divisible by k

Source: Nordic Mathematical Contest 1991 #4

October 5, 2017

Integer Polynomialnumber theoryinteger root

Problem Statement

Let

f(x)

be a polynomial with integer coefficients. We assume that there exists a positive integer

k

and

k

consecutive integers

n, n+1, ... , n+k -1

so that none of the numbers

f(n), f(n+ 1),... , f(n + k - 1)

is divisible by

k

. Show that the zeroes of

f(x)

are not integers.