Existing $a$ and $b$ that given sequence doesn't contain prime number

by jpatrick   Last Updated August 09, 2018 09:20 AM

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.

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