Infinitude of primes via Fermat numbers query

Question

I've just come across the proof of the infinitude of primes using the Fermat numbers. The proof makes use of the relationship $$F_{0}\cdot F_{1}\cdot F_{2}\cdot \ldots \cdot F_{k-1} = F_{k} - 2.$$This is probably a dumb question, but are the Fermat numbers the only series of numbers that have that relationship?

Answer 1

Let $x_0$ be any integer, and set

$$x_k=x_0x_1\cdots x_{k-1} + 2.$$

Then you have a sequence that satisfies this! In fact, as is not too difficult to prove by induction, the values are

$$x_k=(x_0+1)^{2^{k-1}}+1$$

(where $k>0$ - at $k=0$ we have $x_0$ instead).