Inmathematics,Euclid numbers areintegers of the formE_n =p_n# + 1, wherep_n# is thenthprimorial (the product of the firstnprime numbers). They are named after theancient Greek mathematicianEuclid, in connection withEuclid's theorem that there are infinitely many prime numbers.

AEuclid number of the second kind (also calledKummer number) is an integer of the formE_n =p_n# − 1, wherep_n# is thenth primorial.

For example, the first three primes are 2, 3, 5; their product is 30, and the corresponding Euclid number is 31.

The first few Euclid numbers are3,7,31,211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... (sequenceA006862 in theOEIS).

The first few Kummer numbers are 1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ... (sequenceA057588 in theOEIS).

It is sometimes falsely stated thatEuclid's celebrated proof of the infinitude ofprime numbers relied on these numbers.^[1] 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 firstn primes) and reasoned from there to the conclusion that at least one prime exists that is not in that set.^[2]Nevertheless, Euclid's argument, applied to the set of the firstn primes, shows that thenth Euclid number has aprime factor that is not in this set.

Not all Euclid or Kummer numbers are prime.E₆ = 13# + 1 = 30031 = 59 × 509 is the firstcomposite Euclid number, andE₄ = 7# − 1 = 209 = 11 × 19 is the first composite Kummer number.

For alln ≥ 3 the last digit ofE_n is 1, sinceE_n − 1 is divisible by 2 and 5. In other words, since all primorial numbers greater thanE₂ have 2 and 5 as prime factors, they are divisible by 10, thus allE_n ≥ 3 + 1 have a final digit of 1. Likewise, the last digit of every Kummer number is 9.

No Euclid or Kummer numbers areperfect powers.^[3]

It is not known whether there is an infinite number of prime Euclid numbers (primorial primes)^[4] or prime Kummer numbers.^[5]It is also unknown whether every Euclid number is asquarefree number.^[6]

• Euclid–Mullin sequence
• Proof of the infinitude of the primes (Euclid's theorem)

• Caldwell, Chris K.; Gallot, Yves (2002)."On the primality of${\displaystyle n!\pm 1}$ and${\displaystyle 2\times 3\times 5\times \cdots \times p\pm 1}$".Mathematics of Computation.71:442–443.doi:10.1090/S0025-5718-01-01315-1. Retrieved2025-11-07.

• Integer sequences
• Unsolved problems in number theory

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