Can $1+\prod_{i=1}^n a_i$ be prime for all $n$?

Question

This is inspired by Show, by computing several values, that there are composite numbers in this sequence.

Is there an increasing sequence of positive integers $(a_i)|_{i=1}^{\infty} $ such that $1+\prod_{i=1}^n a_i $ is prime for all $n$?

I have no idea how to solve this or whether such a sequence exists or not.

I would be satisfied with an existence proof.

Answer 1

By (a special case of) Dirichlet's Theorem, for any given $m$ there are infinitely many $k$ for which $km+1$ is prime.

So, once you have a sequence $a_1,a_2,\ldots,a_n$ with the desired property, take $m=a_1a_2\cdots a_n$ and choose $a_{n+1}$ to be a value of $k$ such that $k>a_n$ and $km+1$ is prime.