Grab your elliptic curve calculator!

Fortunately we only have to check up to $p_{6}$ , as

$2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1 = 30031 = 59 \times 509$ .

In general, for $n \ge 1$ , $P_{n} = \displaystyle\prod_{k=1}^{n} p_{k} + 1$ is not divisible by $p_{k}$ for $1 \le k \le n$ , but there is nothing implied about its divisibility by $p_{k \gt n}$ . The construct $P_{n}$ is used in Euclid's proof that the set of all primes is infinite. As $n$ increases the likelihood that $P_{n}$ is prime will decrease, as the prime gaps keep generally increasing in size, leading to this follow-up question: Is $P_{n}$ prime for a finite or infinite number of $n$ ?