Chen prime

# Chen prime

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Encyclopedia
A 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...

p is called a Chen prime if p + 2 is either a prime or a product of two primes
Semiprime
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....

. The even number 2p + 2 therefore satisfies Chen's theorem
Chen's theorem
right|thumb|Chen JingrunChen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime . The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original...

.

The Chen primes are named after Chen Jingrun
Chen Jingrun
Chen Jingrun was a Chinese mathematician who made significant contributions to number theory.- Personal life :Chen was the third son in a large family from Fuzhou, Fujian, China. His father was a postal worker. Chen Jingrun graduated from the Mathematics Department of Xiamen University in 1953...

, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

.

The first few Chen primes are
2, 3, 5, 7, 11
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...

, 13
13 (number)
13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....

, 17
17 (number)
17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...

, 19
19 (number)
19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...

, 23
23 (number)
23 is the natural number following 22 and preceding 24.- In mathematics :Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. Twenty-three is also the fifth factorial prime, the third Woodall prime...

, 29
29 (number)
29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...

, 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,...

, 37
37 (number)
37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...

, 41
41 (number)
41 is the natural number following 40 and preceding 42.-In mathematics:Forty-one is the 13th smallest prime number. The next is forty-three, with which it comprises a twin prime...

, 47
47 (number)
47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...

, 53
53 (number)
53 is the natural number following 52 and preceding 54.-In mathematics:Fifty-three is the 16th prime number. It is also an Eisenstein prime....

, 59
59 (number)
59 is the natural number following 58 and preceding 60.-In mathematics:Fifty-nine is the 17th smallest prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary...

, 67
67 (number)
67 is the natural number following 66 and preceding 68. It is an odd number.-In mathematics:Sixty-seven is the 19th prime number , an irregular prime, a lucky prime, the sum of five consecutive primes , and a Heegner number.Since 18! + 1 is divisible by 67 but 67 is not one more than a multiple of...

, 71
71 (number)
71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....

, 83
83 (number)
83 is the natural number following 82 and preceding 84.-In mathematics:Eighty-three is the sum of three consecutive primes as well as the sum of five consecutive primes ....

, 89, 101
101 (number)
101 is the natural number following 100 and preceding 102.It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one"...

, … .

The first few Chen primes that are not the lower member of a pair of twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

s are
2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... .

The first few non-Chen primes are
43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … .

All of the supersingular primes
Supersingular prime (moonshine theory)
In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number.Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups...

are Chen primes.

Rudolf Ondrejka discovered the following 3x3 magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

of nine Chen primes:

 17 89 71 113 59 5 47 29 101

The lower member of a pair of twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

s is a Chen prime, by definition. In August 2009 Twin Prime Search
Twin Prime Search
Twin Prime Search is a distributed computing project that looks for large twin primes. It uses the programs LLR and NewPGen . It was founded on April 13, 2006 by Michael Kwok...

and Primegrid
PrimeGrid
PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...

found the largest known Chen prime, 65516468355 · 2333333 - 1 with 100355 digits.

## Further results

Chen also proved the following generalization: For any even 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...

h, there exist infinitely many primes p such that p + h is either a prime or a semiprime
Semiprime
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....

.

Terence Tao
Terence Tao
Terence Chi-Shen Tao FRS is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory...

and Ben Green
Ben Green (mathematician)
Ben Joseph Green FRS is a British mathematician, specializing in combinatorics and number theory. He is the Herchel Smith Professor of Pure Mathematics at the University of Cambridge.- Early years :...

proved in 2005 that there are infinitely many three-term arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...

s of Chen primes. Recently, Binbin Zhou proved that the Chen primes contain arbitrarily long arithmetic progressions.