mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Euclid numbers are

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 function ...

, 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 Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek ...

mathematician

Euclid Euclid (; ; 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 geometry that largely domina ...

, in connection with

Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proven by Euclid in his work ''Euclid's Elements, Elements''. There are several proof ...

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 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... .

History

It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of

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 A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...

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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

to 3 modulo 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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

. 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 prime In mathematics, a primorial prime is a prime number of the form ''pn''# ± 1, where ''pn''# is the primorial of ''pn'' (i.e. the product of the first ''n'' primes). Primality tests show that: : ''pn''# − 1 is prime for ...

s). It is also unknown whether every Euclid number is a squarefree number.

Generalization

A Euclid number of the second kind (also called Kummer number) is an integer of the form ''E_n'' = ''p''_''n''# − 1, where ''p''_''n''# is the ''n''th 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.

References