Question 54
Let $P(x)$ be a polynomial with integer coefficients such that $P(0) = 0$ and
$$
\text{gcd}(P(0), P(1), P(2), \dots) = 1.
$$
Prove that there are infinitely many positive integers $n$ such that
$$
\text{gcd}(P(n) - P(0), P(n+1) - P(1), P(n+2) - P(2), \dots) = n.
$$
$$
\text{gcd}(P(0), P(1), P(2), \dots) = 1.
$$
Prove that there are infinitely many positive integers $n$ such that
$$
\text{gcd}(P(n) - P(0), P(n+1) - P(1), P(n+2) - P(2), \dots) = n.
$$