This is a part of my answer to this question I think it deserves to be treated separately.

Let $S$ be the set of all primes from $2$ to $p>17$. Let $q$ be the next prime after $p$. Then $q$ divides $rs+1$ for some $r,s\in S$. I wonder if my conjecture is already known. I checked it for all $p<665,000$.

A reformulation of the question is this (motivated by Gjergji Zaimi's comment). Let $p > 17$ be prime. Let $A$ be the set of primes $< p$ considered as a subset of the cyclic multiplicative group $\mathbb{Z}/p\mathbb{Z}^*$. Is it true that the product $A\cdot A$ contains $-1$?

It is interesting to know how large $A\cdot A$ is. This seem to be related to Freiman-type results of Green, Tao and others. Also as Timothy Foo pointed out, perhaps Vinogradov's method of trigonometric sums can apply.