Italian IMO Team Selection Test 1990

Problem:
Let $P(x)=x^3-x+1$. Prove that, for every natural number $n > 1$, the numbers $n,P(n),P(P(n)),\ldots$ taken pairwise are relatively primes.

Solution:
Let $P_k(n)=\underbrace{P(P(\ldots P(n))))}_{k \ \textrm{times}}$. We prove by induction on $k \in \mathbb{N}^*$ that $P_k(n) \equiv 1 \pmod{P_i(n)}$ for every $0 \leq i \leq k-1$, where we set $P_0(n)=n$. In this way it's clear that taken two arbitrary numbers from the sequence, they will be relatively primes. If $k=1$ it is trivial, since $P(n)-n(n^2-1)=1$. Suppose that the statement is true for some $k$. Then, $$P_{k+1}(n)=P^3_{k}(n)-P_{k}(n)+1 \equiv 1 \pmod{P_{k}(n)}$$ and by induction hypothesis, $P_k(n) \equiv 1 \pmod{P_i(n)}$ for every $0 \leq i \leq k-1$, so $P_{k+1}(n) \equiv 1 \pmod{P_i(n)}$ for every $0 \leq i \leq k$ and the desired conclusion follows by the Principle of Mathematical Induction.