Euclid number

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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 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, 9699691, 223092871, 6469693231, 200560490131, ... (sequence A006862 in the OEIS).

Question dropshade.png Unsolved problem in mathematics:
Are there an infinite number of prime Euclid numbers?
(more unsolved problems in mathematics)

It is not known whether there is an infinite number of prime Euclid numbers (primorial primes).

E6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number, demonstrating that not all Euclid numbers are prime.
A Euclid number is congruent to 3 mod 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4. This property implies that no Euclid number can be a square.

For all n ≥ 3 the last digit of En is 1, since En − 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 En ≥ 3+1 have a final digit of 1.

Generalization[edit]

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, the first few such numbers are:

1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ... (sequence A057588 in the OEIS)

Similar to Euclid numbers. It is not known whether there is an infinite number of prime Kummer numbers.

References[edit]

  1. ^ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
  2. ^ "Proposition 20". 

See also[edit]