In mathematics, Euclid numbers are integers of the form , where pn # 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 En = pn # − 1, where pn # 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, ... .
The first few Kummer numbers are 1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ... .
History It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers.Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52. 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. 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 the last digit of En is 1, since 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. Likewise, the last digit of every Kummer number is 9.
No Euclid or Kummer numbers are perfect powers.
Unsolved problems
It is not known whether there is an infinite number of prime Euclid numbers (primorial primes) or prime Kummer numbers. It is also unknown whether every Euclid number is a squarefree number.
See also Euclid–Mullin sequence Proof of the infinitude of the primes (Euclid's theorem)
References
External links