Q:

Is there always at least one prime in intervals of this form?

Take some 4 consecutive primes $p_n,p_{n+1},p_{n+2},p_{n+3}$ where $p_n \geq 5$.
Now form two products: $p_n \cdot p_{n+3}$ and $p_{n+1} \cdot p_{n+2}$.
Is there always at least one prime in the interval $[p_n \cdot p_{n+3},p_{n+1} \cdot p_{n+2}]$ (it does not matter if we have that $p_n \cdot p_{n+3}>p_{n+1} \cdot p_{n+2}$, if that is the case then just reverse the endpoints of the interval)?

A:

It is very likely that there are infinitely many counterexamples. More precisely, by a special case of the prime tuple conjecture (due to Dickson and Hardy-Littlewood), there are infinitely many consecutive primes of the form $p<p+2<p+6<p+8$. The corresponding difference then equals
$$ (p+2)(p+6)-p(p+8)=12, $$
and the requirement is that there is a prime between $p(p+8)$ and $p(p+8)+12$. The expected prime gap size around $p(p+8)$ is $\sim2\log p$, which is much larger than $12$, so we have every reason to believe that the requirement fails for most of the above $p$'s. 

A:

I find about 29,000 cases where this fails with all four primes below 10^8.

