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."
For any two different integers p and q,
and, since (p - q) | (pk - qk) for every integer k>0, Lemma follows.
Assume then that
with all a,b,c, and d different. From Lemma we immediately obtain that
|(d - a) | (f(d) - f(a)) = 3 - 2 = 1||, and|
|(d - b) | (f(d) - f(b)) = 3 - 2 = 1||, and|
|(d - c) | (f(d) - f(c)) = 3 - 2 = 1|
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