Let $P(k)$ be the $k$-th first prime. Is it true that $P(k+1)\leq P(1)\cdot P(2) \cdots P(k)+1$

Question

Let $P(k)$ be the $k$-th first prime. Is it true that $P(k+1)\leq P(1)\cdot P(2)\cdot \dots \cdot P(k)+1$?

Answer 1

Of course. The number on the right is coprime with $P (1),\ldots,P (k) $, and it's divisible by a prime.

Answer 2

If you're interested in a tighter bound, $P(k+1)\le 2P(k)-1$. Sadly, there's no $1$-line proof, unlike the one Martin provided for the original question.