Euclid number

In mathematics, Euclid numbers are integers of the form E_n = p_n# + 1, where p_n# is the nth primorial (the product of the first n prime numbers). They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.

A Euclid number of the second kind (also called Kummer number) is an integer of the form E_n = p_n# − 1, where p_n# is the nth primorial.

Examples

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 are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... (sequence A006862 in the OEIS).

The first few Kummer numbers are 1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ... (sequence A057588 in the OEIS).

History

It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime 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 first n 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 first n primes, shows that the nth Euclid number has a prime factor that is not in this set.

Properties

Not all Euclid or Kummer numbers are prime. E₆ = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number, and E₄ = 7# − 1 = 209 = 11 × 19 is the first composite Kummer number.

For all n ≥ 3 the last digit of E_n is 1, since E_n − 1 is divisible by 2 and 5. In other words, since all primorial numbers greater than E₂ have 2 and 5 as prime factors, they are divisible by 10, thus all E_n ≥ 3 + 1 have a final digit of 1. Likewise, the last digit of every Kummer number is 9.

No Euclid or Kummer numbers are perfect powers.^[3]

Unsolved problems

Unsolved problem in mathematics

Are there an infinite number of prime Euclid numbers?

References

^ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
^ "Proposition 20".
^ Lord, Nick (2014). "Euclid numbers are free from powers". The Mathematical Gazette. 98 (543): 482–483. doi:10.1017/S0025557200008184.
^ Sloane, N. J. A. (ed.). "Sequence A006862 (Euclid numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A125549 (Composite Kummer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. ISBN 9780201529890.

External links

Caldwell, Chris K.; Gallot, Yves (2002). "On the primality of $n!\pm 1$ and $2\times 3\times 5\times \dotsb \times p\pm 1$ ". Mathematics of Computation. 71: 442–443. doi:10.1090/S0025-5718-01-01315-1. Retrieved 2025-11-07.

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

[2] "Proposition 20".

[3] Lord, Nick (2014). "Euclid numbers are free from powers". The Mathematical Gazette. 98 (543): 482–483. doi:10.1017/S0025557200008184.

[4] Sloane, N. J. A. (ed.). "Sequence A006862 (Euclid numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[5] Sloane, N. J. A. (ed.). "Sequence A125549 (Composite Kummer numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[6] Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. ISBN 9780201529890.

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