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In mathematics, Euclid numbers are integers of the form E_n = p_n# + 1, where p_n# is the nth primorial, i.e. the product of the first n primes. They are named after the ancient Greek mathematician Euclid.

It is sometimes falsely stated[1] that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers. Euclid did not begin with the assumption that the set of all primes is finite. Rather, he said: consider any finite set of primes (he did not assume that it contained only the first n primes, e.g. it could have been {3, 41, 53}) and reasoned from there to the conclusion that at least one prime exists that is not in that set.[2]

The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511 (sequence A006862 in OEIS).

List of unsolved problems in mathematics
Are there an infinite number of prime Euclid numbers?

It is not known whether or not there are an infinite number of prime Euclid numbers.

E₆ = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number, demonstrating that not all Euclid numbers are prime.

A Euclid number can not be a square. This is implied by Euclid numbers always being congruent to 3 mod 4.

For all n ≥ 3 the last digit of E_n is 1, since E_n − 1 is divisible by 2 and 5.

References

Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.

"Proposition 20".