In mathematics, Euclid numbers are integers of the form En = pn # + 1, where pn # is the nth primorial, i.e. the product of the first n prime numbers.
[1] Euclid did not begin with the assumption that the set of all primes is finite.
[2] Nevertheless, Euclid's argument, applied to the set of the first n primes, shows that the nth Euclid number has a prime factor that is not in this set.
In other words, since all primorial numbers greater than E2 have 2 and 5 as prime factors, they are divisible by 10, thus all En ≥ 3 + 1 have a final digit of 1.
[4] A Euclid number of the second kind (also called Kummer number) is an integer of the form En = pn # − 1, where pn # is the nth primorial.