In the following we'll assume f(x) is a polynomial with integral coefficients:

Also, the vertical bar (the pipe symbol) is used to indicate divisibility: a|b is a shorthand for "a divides b."

Lemma

For any two different integers p and q, f(p) - f(q) is divisible by p - q:

Proof

Indeed,

and, since (p - q) | (p^k - q^k) for every integer k>0, Lemma follows.

Assume then that

with all a,b,c, and d different. From Lemma we immediately obtain that

Thus differences d - a, d - b, d - c all divide 1. But 1 has only two divisors: 1 and -1. Therefore, by the Pigeonhole Principle, two of the differences coincide. Which contradicts our assumption that the numbers a, b, c are all different.