2. Let $f(x)=a x^{2}+b x+c$ be a quadratic trinomial with integer coefficients. If integers $m, n$ satisfy $f(m)-f(n)=1$, then $|m-n|=$ $\qquad$ .
2. 1 .

From
$$
f(m)-f(n)=a\left(m^{2}-n^{2}\right)+b(m-n)=(m-n)(a(m+n)+b)=1,
$$

we get $(m-n)(a m+a n+b)=1$. Since $m-n, a m+a n+b$ are both integers, it must be that $|m-n|=1$.