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)?



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


