Prove that there is no polynomial $$P(x)=a_nx^n+a_{n-1}x^{n-1}+ \dots+ a_0$$ with integer coefficients and of degree at least $1$ with the property that $P(0), P(1), P(2), \dots$ are all prime numbers.
How should one approach this? Contradiction seems plausible if we would assume it we would get that $P(0), P(1), P(2) \dots$ would all equal some primes. Also from $P(0) = q$, where $q$ is some prime we would get that $a_0=q.$ From here on I don’t quite know how to continue… Any hints would be appreciated.
Best Answer
The key point here is that $a-b|P(a)-P(b)$ (if you haven’t seen this already you should try proving it, it’s a nice exercise).
From here, if $P(n) = p$, then $p$ divides $P(n+kp)$ for all positive $k$. But since all of these are prime, we get that $P(n+kp)=p$, and so $P$ takes the same value infinitely many times, and hence is constant, contradicting the degree condition.