astral shuttle
Can we prove that there always exist a prime in the range $\left(\prod^n_{i=1}p_i-p_{n+1}^2 , \prod^n_{i=1}p_i\right)$ where $p_i$ is the i-th prime number and $n>1$?
I'll call $P_n=\prod^n_{i=1}p_i$ for simplicity's sake.
By Bertrand–Chebyshev theorem there exists a prime $p\in(P_n-p^2_{n+1},2(P_n-p^2_{n+1}))$. But then checking to see if: $$(P_n-p^2_{n+1},2(P_n-p^2_{n+1}))\subset(P_n-p^2_{n+1} , P_n)\implies \ 2(P_n-p^2_{n+1})\leq P_n\iff P_n-2p^2_{n+1}\leq 0$$ Which holds for $n\leq4$ but then the values become positive.
Could anyone help me with this please?
glad harbor