Let $f\in \mathbb{Q} [x]$ be a polynomial, and $a_0 = a$ be an arbitrary integer. Let us define a sequence $\{a_n \} $ by the recurrence relationship : $$a_n = f(a_{n-1} ). $$ I want to show that $a_n $ cannot always be a prime number, with $\{a_n \}$ being pairwise distinct. I am pretty sure that this is a very well known fact, but I cannot easily find this.

Note : I posted this question also in SE Math.

http://math.stackexchange.com/questions/1568513/non-existence-of-a-prime-generating-polynomial-recurrence-relation

Let $f\in \mathbb{Q} [x]$ be a polynomial, and $a_0 = a$ be an arbitrary integer. Let us define a sequence $\{a_n \} $ by the recurrence relationship : $$a_n = f(a_{n-1} ). $$ I want to show that $a_n $ cannot always be a prime number, with $\{a_n \}$ being pairwise distinct. I am pretty sure that this is a very well known fact, but I cannot easily find this.

Note : I posted this question also in SE Math.

https://math.stackexchange.com/questions/1568513/non-existence-of-a-prime-generating-polynomial-recurrence-relation