Chen prime

Chen prime
Named after	Chen Jingrun
Publication year	1973[1]
Author of publication	Chen, J. R.
First terms	2, 3, 5, 7, 11, 13
OEIS index	A109611; Chen primes: primes p such that p + 2 is either a prime or a semiprime;

A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem.

The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture as the lower member of a pair of twin primes is by definition a Chen prime.

The first few Chen primes are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, … (sequence A109611 in the OEIS).

The first few Chen primes that are not the lower member of a pair of twin primes are

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... (sequence A063637 in the OEIS).

The first few non-Chen primes are

43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … (sequence A102540 in the OEIS).

All of the supersingular primes are Chen primes.

Rudolf Ondrejka discovered the following 3x3 magic square of nine Chen primes:[2]

17	89	71
113	59	5
47	29	101

2996863034895 × 2^1290000 − 1, with 388342 decimal digits, is the largest known Chen prime as of March 2018.

The sum of the reciprocals of Chen primes converges.

Further results

Chen also proved the following generalization: For any even integer h, there exist infinitely many primes p such that p + h is either a prime or a semiprime.

Green and Tao showed that the Chen primes contain infinitely many arithmetic progressions of length 3.[3] Binbin Zhou generalised this result by showing that the Chen primes contain arbitrarily long arithmetic progressions.[4]

Notes

1.^{^} Chen primes were first described by Yuan, W. On the Representation of Large Even Integers as a Sum of a Product of at Most 3 Primes and a Product of at Most 4 Primes, Scienca Sinica 16, 157-176, 1973.

References

Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 17: 385–386.
Prime Curios! page on 59
Ben Green and Terrence Tao, Restriction theory of the Selberg sieve, with applications, Journal de Théorie des Nombres de Bordeaux 18 (2006), pp. 147–182.
Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138:4 (2009), pp. 301–315.

External links

The Prime Pages
Green, Ben; Tao, Terence (2006). "Restriction theory of the Selberg sieve, with applications". Journal de théorie des nombres de Bordeaux. 18 (1): 147–182. arXiv:math.NT/0405581. doi:10.5802/jtnb.538.
Weisstein, Eric W. "Chen Prime". MathWorld.
Zhou, Binbin (2009). "The Chen primes contain arbitrarily long arithmetic progressions". Acta Arith. 138 (4): 301–315. Bibcode:2009AcAri.138..301Z. doi:10.4064/aa138-4-1.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 17: 385–386.

[2] Prime Curios! page on 59

[3] Ben Green and Terrence Tao, Restriction theory of the Selberg sieve, with applications, Journal de Théorie des Nombres de Bordeaux 18 (2006), pp. 147–182.

[4] Binbin Zhou, The Chen primes contain arbitrarily long arithmetic progressions, Acta Arithmetica 138:4 (2009), pp. 301–315.

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers