Chen prime number

In number theory , a Chen prime number is a prime number for which the following applies: ${\displaystyle p\in \mathbb {P} }$

${\displaystyle p+2}$is either a prime number or a product of two prime numbers (a so-called semi- prime number ).

These prime numbers were named by Ben Green and Terence Tao in memory of the Chinese mathematician Chen Jingrun .

history

One of the best known unsolved problems in mathematics is Goldbach's conjecture from 1742, which states that every even number (greater than 2) is the sum of two prime numbers. Mathematicians have been grappling with this for hundreds of years, and David Hilbert made this assumption one of the 23 most important mathematical problems in 1900 . 14 of these problems have now been solved, six problems have been partially solved and only three problems have not been solved. Goldbach's Conjecture is one of these remaining three problems (Hilbert's eighth problem). In 1966, the Chinese mathematician Chen Jingrun proved Chen's theorem named after him ( but not published until 1973 because of the Chinese Cultural Revolution ), which says that every sufficiently large even number is the sum of two prime numbers or a prime number and a semi- prime number ( i.e. one Number with two prime factors) can be written. This theorem is the best approximation to date of the Goldbach conjecture mentioned above . If now is a Chen prime, then the even number fulfills the requirements of Chen's theorem (it can be represented in the form as the sum of a prime and a prime or semi- prime number ). With Chen's theorem it was thus proved that there are infinitely many such Chen prime numbers. In addition, if the mathematical conjecture that there are infinitely many twins of prime numbers is proven at some point (many mathematicians have failed on this question for centuries), one would have found an alternative proof that there are infinitely many Chen prime numbers. ${\displaystyle p}$ ${\displaystyle 2p+2}$ ${\displaystyle 2p+2=p+(p+2)}$ ${\displaystyle p}$ ${\displaystyle p+2}$

Examples

• The prime number is a Chen prime number because it is also a prime number. ${\displaystyle p=41}$ ${\displaystyle p+2=43}$
• The prime number is a Chen prime number because it is a semi -prime number, i.e. it has exactly two prime divisors. ${\displaystyle p=83}$ ${\displaystyle p+2=85=5\cdot 17}$
• The prime number is a Chen prime number because it is a semi -prime number, i.e. it has exactly two prime divisors. ${\displaystyle p=47}$ ${\displaystyle p+2=49=7^{2}}$
• The prime is not a Chen prime because it is not a semi -prime because it has three prime divisors. ${\displaystyle p=43}$ ${\displaystyle p+2=45=5\cdot 3^{2}}$
• The first Chen primes are the following:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409, ... (follow A109611 in OEIS )

The first prime numbers that are not Chen prime numbers can be found in the following list:

43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, 271, 277, 283, 313, 331, 349, 367, 373, 383, 397, 421, 433, 439, 457, 463, 523, 547, 593, 601, 607, 613, 619, 643, 661, 673, 691, 709, 727, 733, 739, 757, 773, 823, 853, 859, 883, 907, 929, 967, 997,… (sequence A102540 in OEIS )

• The first Chen primes that are not the minor part of a prime twin (that is, for which is a semi -prime ) are as follows: ${\displaystyle p+2}$

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677, ... (sequence A063637 in OEIS )

• The largest Chen prime to date (as of July 10, 2018) is the following:

${\displaystyle p=2996863034895\cdot 2^{1290000}-1}$

It has 388,342 positions and was discovered by Tom Greer on September 14, 2016. It is the smaller part of the largest known prime twin to date . ${\displaystyle (p,p+2)}$

properties

• There are an infinite number of Chen prime numbers.
• If a Chen prime is part of a prime twin, it is the smaller part of the prime twin.

Proof:

Since in the case of Chen prime numbers the number must be a prime number or a semi -prime number, the Chen prime number is always the smaller part of the prime twin. ${\displaystyle p}$ ${\displaystyle p+2}$ ${\displaystyle (p,p+2)}$ ${\displaystyle p}$ ${\displaystyle \Box }$

• There are magic squares made up of only Chen prime numbers. The magic square with the smallest Chen prime numbers is the following:

17th	89	71
113	59	5
47	29	101

It was discovered by Rudolf Ondrejka.

• There are infinitely many Chen prime numbers in arithmetic sequences of length 3 (the difference between two sequence terms is constant).

(The above sentence was proven by Ben Green and Terence Tao in 2005.)

Example:

The largest sequence of prime numbers of length 3 so far (as of July 10, 2018), which consists exclusively of Chen prime numbers, is the following:

${\displaystyle p_{n}=(3850324118+892819689\cdot n)\cdot 2411\#+1)\cdot (4787\#+1)-2}$ With ${\displaystyle n=0,1,2}$

Here is the product of all prime numbers up to and including ( prime faculty ). ${\displaystyle p\#=2\cdot 3\cdot 5\cdot 7\cdot \ldots \cdot p}$ ${\displaystyle p}$

These three numbers all have 3074 digits each. For all three prime numbers there is a semi- prime number (i.e. they have exactly two prime factors). Because these three prime numbers form an arithmetic sequence, from is the same distance as from (so it is ). ${\displaystyle p_{n}}$ ${\displaystyle p_{n}+2}$ ${\displaystyle p_{0}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{2}-p_{1}=p_{1}-p_{0}}$

• There are an infinite number of Chen prime numbers in arithmetic sequences of any length.

(This theorem was proven by Binbin Zhou in 2009 and is an improvement on the previous theorem.)

generalization

A generalized Chen prime is a prime for which the following applies: ${\displaystyle p\in \mathbb {P} }$

${\displaystyle p+k}$with a natural number is either a prime number or a product of two prime numbers (a so-called semi- prime number ). ${\displaystyle k\in \mathbb {N} }$

properties

• There are an infinite number of generalized Chen prime numbers.

useful information

It looks like there are more Chen prime numbers than non-Chen prime numbers (see prime number lists above ). Below there are 20 Chen prime numbers, but only 5 non-Chen prime numbers. The number of Chen prime numbers also predominates (115), because there are only 53 non-Chen prime numbers among them. However, this ratio changes as the size of the prime numbers increases. Below there are 986 Chen prime numbers and also 986 non-Chen prime numbers. After that, the non-Chen primes predominate. This fact is another example of the fact that one does not have to agree with regularities that one believes to be noticed with small numbers for all other, larger numbers. With the Goldbach conjecture already mentioned above (every even number (greater than 2) can be represented as the sum of two prime numbers), the conjecture was checked with computer technology up to the order of magnitude (i.e. up to a trillion ) (as of December 30, 2015) and for found to be correct, therefore hardly any mathematician believes that this conjecture turns out to be wrong (for the proof of this conjecture even a prize money of one million dollars was awarded, but nobody could prove this conjecture), but nobody can rule out that does not at some point open up a counterexample, be it of the order of magnitude or even higher. ${\displaystyle n=100}$ ${\displaystyle n=1000}$ ${\displaystyle n=17107}$ ${\displaystyle n=10^{18}}$ ${\displaystyle n=10^{100000}}$

Individual evidence

1. ↑ ^{a b} Chen prime . In: PlanetMath . (English)
2. ↑ Neil Sloane : Chen primes: primes p such that p + 2 is either a prime or a semiprime - Comments. OEIS , accessed July 10, 2018 . 
3. ↑ ^{a b c} Chen Jingrun : On the representation of a larger even integer as the sum of a prime and the product of at most two primes. In: Scientia Sinica 16 (2). 1973, pp. 157–176 , accessed on July 10, 2018 (English). 
4. ↑ ^{a b} Konstantin Fackeldey: The Goldbach conjecture and its previous attempts at a solution. Freie Universität Berlin , 2002, p. 26 , accessed on July 10, 2018 (German). 
5. ↑ Chris K. Caldwell: The Top Twenty: Twin Primes. Prime Pages, accessed July 10, 2018 . 
6. ↑ 2996863034895 • 2 ^1290000 - 1 on Prime Pages
7. ↑ Ben Green , Terence Tao : Restriction theory of the Selberg sieve, with applications. Journal de Théorie des Nombres , 2005, pp. 1–36 , accessed on July 10, 2018 . 
8. ↑ Eric W. Weisstein : Chen Prime . In: MathWorld (English).
9. ↑ Binbin Zhou : The Chen primes contain arbitrarily long arithmetic progressions. Acta Arithmetica 138 (4), 2009, pp. 301-315 , accessed July 10, 2018 . 
10. ↑ Tomás Oliveira e Silva: Goldbach conjecture verification. 2015, accessed July 10, 2018 . 

Web links

• Eric W. Weisstein : Chen Prime . In: MathWorld (English).
• Chen prime . In: PlanetMath . (English)
• Chris K. Caldwell: Chen Prime. Prime Pages, accessed July 10, 2018 .