Let $p(x)$ be a polynomial with integer coefficients. Suppose there exist $3$ different integers $a$, $b$, and $c$, such that $$p(a) = p(b) = p(c) = 3.$$ Now using the fact that $x−y$ divides $p(x)−p(y)$ for any two $x,y\in\mathbb Z$ and the pigeonhole
principle to show that there is no integer $d$ with $p(d) = 4$.
How do you apply the pigeonhole principle here? Thank you in advance.
Best Answer
Suppose otherwise. Then, using $u-v\mid p(u)-p(v)$, $$d-a,d-b,d-c$$ are divisors of $4-3=1$. This is only possible if these numbers are $1$ or $-1$. Then, by Pigeonhole, two of these numbers are equal, contradicting the fact that $a$, $b$, $c$ are distinct. $\blacksquare$