- Euclid number
In
mathematics , Euclid numbers areinteger s of the form "E""n" = "p""n"# + 1, where "p""n"# is theprimorial of "p""n" which is the "n"th prime. They are named after theancient Greek mathematician Euclid .It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of
prime number s relied on these numbers. In fact, 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 it contained just 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. [cite web|last= |first= |authorlink= |coauthors= |title=Proposition 20 |work= |publisher= |date= |url=http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html |format= |doi= |accessdate= |quote = ]The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511 OEIS|id=A006862.
It is not known whether or not there are an infinite number of prime Euclid numbers.
"E""6" = 13# + 1 = 30031 = 59 x 509 is the first composite Euclid number, demonstrating that not all Euclid numbers are prime.
A Euclid number can not be aperfect power .For all "n" ≥ 3 the last digit of "E""n" is 1, since "E""n"−1 is divisible by 2 and 5.
References
See also
*
Euclid-Mullin sequence
* Proof of the infinitude of the primes (Euclid's theorem)
*Primorial prime
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