DylanN wrote: Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$. If $P(x)-P(y)=d$ has infinitely many solutions, then infinitely many of these solutions have either $|x|$, either $|y|$ as great as we want. But when $x$ or $y$, WLOG $|x|$ is towards $\pm\infty$, we clearly have $|P(x)-P(y)|\ge\min(|P(x)-P(x-1)|,|P(x+1)-P(x)|)$ and so is unbounded, impossible So degree is $0$ or $1$ and we immediately get the result : Either $f(x)=a$ and $d=0$ Either $f(x)=ax+b$ and $d=ka$

What about solutions like $p(x) = x^2$ for $d = 0$?

DylanN wrote: What about solutions like $p(x) = x^2$ for $d = 0$? Hummmmpf .... It seems you are right