The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (OEIS A006862; Tietze 1965,
p. 19).
The indices
of the first few prime Euclid numbers are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643,
... (OEIS A014545), so the first few Euclid
primes (commonly known as primorial primes) are
3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239).
For example,
is a prime with 63142 digits
(OEIS A014545). It is not known if there are
an infinite number of prime
Euclid numbers (Guy 1994, Ribenboim 1996).
The largest factors of for , 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS
A002585).
is defined by
is the 
th
prime and 
is the primorial. The second
of Euclid's theorems states that the number of
primes is infinite.
Its proof can be accomplished using the Euclid numbers.
of the first few prime Euclid numbers 
are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643,
... (OEIS A014545), so the first few Euclid
primes (commonly known as primorial primes) are
3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239).
For example, 
is a prime with 63142 digits
(OEIS A014545). It is not known if there are
an infinite number of prime
Euclid numbers (Guy 1994, Ribenboim 1996).
for 
, 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS
A002585).
