Is this estimate $p_{n+1}^2 < p_1 \cdot p_2 \cdot ... p_n$ true for prime numbers?

Question

I think to remember that if $p_1, ..., p_n$ are the first $n$ prime numbers, then $$ p_{n+1}^2 < p_1 \cdot p_2 \cdot ... p_n $$ for all $n \ge 4$, but I can't find a reference for it. If true, is it obvious (something like Euclid's trick) or more like Bertrand's postulate, etc.?

Answer 1

Yes.

By Berty, $p_{n+1} < 2p_n < 4p_{n-1} < 8p_{n-2} ...$ so $p_{n+1}^3 < 64p_{n}p_{n-1}p_{n-2}$ so $p_{n+1}^2 < p_{n}p_{n-1}p_{ n-2}$ for $p_{n+1} > 64$.