Does exist $a$ and $b$, which are coprime positive integers that sequence defined below contains only composite numbers: $$x_0=a, \ x_{n+1}=b+\prod_{i=0}^n x_i ?$$
I suppose that it doesn't exist, because proving that it exist looks veeery messy. It is easy to observe that $x_n \equiv b \ (\text{mod } a)$ and $x_n \equiv a^{2^{n-1}} \ (\text{mod } a)$ for $n \geq 2$. From that I can compute that $x_n \equiv a^{2^n}+b \ (\text{mod } ab)$ for $n\geq 2.$ Then I tried use Dirichlet theorem about arithmetic progression, but it doesn't look good aproach. It is also easy to notice that terms of this sequence are pair coprime, but it is trivial to construct sequence builded by composite numbers, but having this property.
Can you provide me ONLY at this moment a hint? I would like to solve it myself, but I'm stucked. No solution, please.
The non-linearity strongly suggests that you won't be able to prove that the sequence must contain a prime.
Thus, the better guess is that it is possible for the sequence to have all terms composite.
Identically, for $n\ge 0$, we have $$x_{n+2}=b+bt+t^2$$ where $$t=\prod_{i=0}^n x_i$$ hence, choosing $b=4$, it follows that for any positive integer $a$, the terms $x_2,x_3,x_4,...\;$are all composite (why?).
It remains to find an odd positive integer $a$ so that $x_0,x_1$ are also composite.