Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers
(1)
(2)
known as Euclid numbers, where is the th prime and is the primorial.
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).
The largest known Euclid number is , and 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 the 
th prime and 
is the primorial.
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).
The largest known Euclid number is 
, and 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).
