Primality of Euclid Numbers

Open Question

Consider the sequence of Euclid numbers:

The sequence of Euclid numbers begins as follows:

\(\ds E_0 \ \ \)

\(\ds = p_0\# + 1\)

\(=\)

\(\ds 1 + 1\)

\(\ds = 2\)

\(\ds E_1 \ \ \)

\(\ds = p_1\# + 1\)

\(=\)

\(\ds 2 + 1\)

\(\ds = 3\)

\(\ds E_2 \ \ \)

\(\ds = p_2\# + 1\)

\(=\)

\(\ds 2 \times 3 + 1\)

\(\ds = 7\)

\(\ds E_3 \ \ \)

\(\ds = p_3\# + 1\)

\(=\)

\(\ds 2 \times 3 \times 5 + 1\)

\(\ds = 31\)

\(\ds E_4 \ \ \)

\(\ds = p_4\# + 1\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 + 1\)

\(\ds = 211\)

\(\ds E_5 \ \ \)

\(\ds = p_5\# + 1\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 \times 11 + 1\)

\(\ds = 2311\)

\(\ds E_6 \ \ \)

\(\ds = p_6\# + 1\)

\(=\)

\(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1\)

\(\ds = 30031\)

It is not known whether there is an infinite number of:

Euclid primes

non-prime Euclid numbers.

Also see

• Definition:Euclid Prime/Sequence:

The sequence of Euclid primes begins:

$2, 3, 7, 31, 211, 2311, 200 \, 560 \, 490 \, 131, \ldots$