Polynomial problem involving divisibility, prime numbers, monotony Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio $r>0$ such that the value of function $f$ in every term of the progression is a prime number.
I believe that the solution includes a reductio ad absurdum, but I don't know how to solve it.

If progression is $a+nb$,  and $f(a)=p$, then $f(a+pnb)\equiv 0\pmod p$ and absolute value is more then $p$ for large $n$.
