A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes. In 1966 Chen Jingrun proved that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... 1966 (MCMLXVI) was a common year starting on Saturday (the link is to a full 1966 calendar). ... Chen Jingrun (ch. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...

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 OEIS) “II” redirects here. ... This article discusses the number three. ... Look up five in Wiktionary, the free dictionary. ... Seven Days of Creation - 1765 book, title page Seven (7) is the sixth successor of one (1). ... 11 (eleven) is the natural number following 10 and preceding 12. ... 13 (thirteen) is the natural number following 12 and preceding 14. ... 17 (seventeen) is the natural number following 16 and preceding 18. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... This article is about the number 23. ... 29 (twenty-nine) is the natural number following 28 and preceding 30. ... 31 (thirty-one) is the natural number following 30 and preceding 32. ... 37 (thirty-seven) is the natural number following 36 and preceding 38. ... 41 (forty-one) is the natural number following 40 and preceding 42. ... 47 (forty-seven) is the natural number following 46 and followed by 48. ... 53 (fifty-three) is the natural number following 52 and preceding 54. ... 59 (fifty-nine) is the natural number following 58 and preceding 60. ... 67 (sixty-seven) is the natural number following 66 and preceding 68. ... 71 is the natural number following 70 and preceding 72. ... 83 (eighty-three) is the natural number following 82 and preceding 84. ... 89 (eighty-nine) is the natural number following 88 and preceding 90. ... 101 (one hundred [and] one) is the natural number following 100 and preceding 102. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

The first few non-Chen primes are

43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241 (sequence A102540 in OEIS) 43 (forty-three) is the natural number following 42 and preceding 44. ... 61 (sixty-one) is the natural number following 60 and preceding 62. ... 73 (seventy-three) is the natural number following 72 and preceding 74. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Note that all of the supersingular primes are Chen primes. In mathematics, a supersingular prime is a certain kind of prime number. ...

Rudolf Ondrejka (1928-2001) discovered the following 3x3 magic square of nine Chen primes^[1]: It has been suggested that Date magic square be merged into this article or section. ...

In October 2005 Micha Fleuren and PrimeForm e-group found the largest known Chen prime, (1284991359 · 2⁹⁸³⁰⁵ + 1) · (96060285 · 2¹³⁵¹⁷⁰ + 1) − 2 with 70301 digits. 2005 (MMV) was a common year starting on Saturday of the Gregorian calendar. ...

The lower member of a pair of twin primes is a Chen prime, by definition. As of January 16, 2007, the largest known twin prime is 2003663613 · 2¹⁹⁵⁰⁰⁰ ± 1. A twin prime is a prime number that differs from another prime number by two. ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ...

Terence Tao and Ben Green proved in 2005 that there are infinitely many three-term arithmetic progressions of Chen primes. Terence Chi-Shen Tao ( 陶哲軒)(born 1975) is an Australian mathematician working primarily on harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. ... Ben Joseph Green (born February 27, 1977, Bristol, United Kingdom) is a British mathematician, specializing in combinatorics and number theory. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ...

Further Results

Chen also proved the following.

For any even integer h, there exists infinitely many primes p such that p + h is either a prime or a product of two primes.. This article or section is not written in the formal tone expected of an encyclopedia article. ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ...

References

External links

• The Prime Pages
• Ben Green, Terence Tao, Restriction theory of the Selberg sieve, with applications
• Eric W. Weisstein, Chen Prime at MathWorld.