Euclid number
Template:Distinguish Template:Short description In mathematics, Euclid numbers are integers of the form Template:Nowrap, 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, ... Template:OEIS.
The first few Kummer numbers are 1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ... Template: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 Template:Nowrap the last digit of En is 1, since Template:Nowrap 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.[3]
Unsolved problems
Template:Unsolved Template:Unsolved 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 a squarefree number.[6] Template:Unsolved
See also
- Euclid–Mullin sequence
- Proof of the infinitude of the primes (Euclid's theorem)
References
External links
Template:Prime number classes Template:Classes of natural numbers
- ↑ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
- ↑ Template:Cite web
- ↑ Template:Cite journal
- ↑ Template:Cite OEIS
- ↑ Template:Cite OEIS
- ↑ Template:Cite book