Euclid number
Encyclopedia
In mathematics
, Euclid numbers are integer
s of the form En = pn# + 1, where pn# is the primorial
of pn which is the nth prime. They are named after the ancient 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.
The first few Euclid numbers are 3, 7, 31
, 211
, 2311, 30031, 510511 .
It is not known whether or not there are an infinite number of prime Euclid numbers.
E6 = 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 a square
. This is because Euclid numbers are always congruent to 3 mod 4.
For all n ≥ 3 the last digit of En is 1, since En − 1 is divisible by 2 and 5.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Euclid numbers are integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s of the form En = pn# + 1, where pn# is the primorial
Primorial
In mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
of pn which is the nth prime. They are named after the ancient Greek
Ancient Greek
Ancient Greek is the stage of the Greek language in the periods spanning the times c. 9th–6th centuries BC, , c. 5th–4th centuries BC , and the c. 3rd century BC – 6th century AD of ancient Greece and the ancient world; being predated in the 2nd millennium BC by Mycenaean Greek...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
.
It is sometimes falsely stated that Euclid's celebrated proof
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...
of the infinitude of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
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.
The first few Euclid numbers are 3, 7, 31
31 (number)
31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 211
211 (number)
211 is the natural number between 210 and 212. It is also a prime number.-In mathematics:211 is an odd number.211 is a primorial prime, sum of three consecutive primes , Chen prime, centered decagonal prime, and self prime....
, 2311, 30031, 510511 .
It is not known whether or not there are an infinite number of prime Euclid numbers.
E6 = 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 a square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
. This is because Euclid numbers are always congruent to 3 mod 4.
For all n ≥ 3 the last digit of En is 1, since En − 1 is divisible by 2 and 5.
See also
- Euclid–Mullin sequence
- Proof of the infinitude of the primes (Euclid's theorem)
- Primorial primePrimorial primeIn mathematics, primorial primes are prime numbers of the form pn# ± 1, where:The first few primorial primes are, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid project....