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In mathematics, Euclid numbers are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s of the form , where ''p''''n''# is the ''n''th
primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the functi ...
, i.e. the product of the first ''n''
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. They are named after the ancient Greek
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
, in connection with
Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work '' Elements''. There are several proofs of the theorem. Euclid's proof Euclid off ...
that there are infinitely many prime numbers.


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, ... .


History

It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s 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 ) 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 ''n''th Euclid number has a prime factor that is not in this set.


Properties

Not all Euclid numbers are prime. ''E''''6'' = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number. Every 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 In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
. For all the last digit of ''E''''n'' is 1, since is divisible by 2 and 5. In other words, since all primorial numbers greater than ''E''2 have 2 and 5 as prime factors, they are divisible by 10, thus all ''E''''n'' ≥ 3+1 have a final digit of 1.


Unsolved problems

It is not known whether there is an infinite number of prime Euclid numbers ( primorial primes). It is also unknown whether every Euclid number is a
squarefree number In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...
.


Generalization

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, ... As with the Euclid numbers, it is not known whether there are infinitely many prime Kummer numbers. The first of these numbers to be composite is
209 Year 209 ( CCIX) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Commodus and Lollianus (or, less frequently, year 962 ''Ab urbe condi ...
.


See also

*
Euclid–Mullin sequence The Euclid–Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements. They are named after the ancient Greek mathematician Euclid, because the ...
* Proof of the infinitude of the primes (Euclid's theorem)


References

{{Classes of natural numbers Integer sequences Unsolved problems in number theory