Prime cousin

In mathematics , prime numbers with a difference of 4 are called prime cousins . For example, the numbers 13 and 17 are prime cousins ​​because one number is 4 smaller than the other (or the other is 4 larger than one).

Prime cousins ​​have the form . The prime cousins ​​under 1000 are ${\displaystyle (p,p+4)}$

(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97 , 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281 ), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487 , 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827 ), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971).

(Follow A023200 in OEIS ) and (Follow A046132 in OEIS )

properties

The only prime that belongs to two pairs of prime cousins ​​is 7. One of the numbers or is always divisible by 3, so this is the only case where the triple consists of three prime numbers. ${\displaystyle p,p+4}$ ${\displaystyle p+8}$ ${\displaystyle p=3}$ ${\displaystyle (p,p+4,p+8)}$

In May 2009, Ken Davis discovered the two currently largest prime cousins ​​with 11594 digits. Of the prime cousins, the first is prime ${\displaystyle (p,p+4)}$ ${\displaystyle p}$

${\displaystyle p=(311778476\cdot 587502\cdot 9001\#\cdot (587502\cdot 9001\#+1)+210)\cdot {\frac {587502\cdot 9001\#-1}{35}}+1}$

It is a Primfakultät , d. H. the product of all prime numbers . ${\displaystyle 9001\#=2\cdot 3\cdot 5\cdot \dotsc \cdot 9001}$ ${\displaystyle \leq 9001}$

The greatest known prime cousins could be the following two: ${\displaystyle (p,p+4)}$

${\displaystyle p:=474435381\cdot 2^{98394}-5}$

${\displaystyle p+4=474435381\cdot 2^{98394}-1}$

They each have 29,629 positions and were discovered in November 2012 by Michael Angel, Paul Jobling and Dirk Augustin. The second of these two numbers,, has now been verified as a prime number, but there is currently no known prime number test that could simply determine whether the first number ,, is prime. is a PRP number ( probable prime ), so it is very likely a prime number because it fulfills conditions which all prime numbers have but which most composite numbers do not fulfill. ${\displaystyle p+4}$ ${\displaystyle p}$ ${\displaystyle p}$

It follows from the first Hardy-Littlewood conjecture that prime cousins have the same asymptotic density as prime twins . An analogy to Brun's constant for prime twins can also be defined for prime cousins. It is called Brun's constant for prime cousins and is the result of the convergent sum

${\displaystyle B_{4}=\left({\frac {1}{7}}+{\frac {1}{11}}\right)+\left({\frac {1}{13}}+{\frac {1}{17}}\right)+\left({\frac {1}{19}}+{\frac {1}{23}}\right)+\cdots .}$

The first prime number cousin pair (3, 7) is omitted.

If you insert all prime number cousins ​​to , Marek Wolf showed in 1996 that: ${\displaystyle 2^{42}}$

${\displaystyle B_{4}\approx 1{,}1970449}$(Follow A194098 in OEIS )

This constant must not be confused with Brun's constant for prime quadruplets, which is also referred to as, but gives a different value. ${\displaystyle B_{4}}$

literature

• David Wells: Prime Numbers: The Most Mysterious Figures in Math . John Wiley & Sons, 2011, ISBN 1-118-04571-8 , pp. 33 .
• Benjamin Fine, Gerhard Rosenberger: Number theory: an introduction via the distribution of primes . Birkhäuser, 2007, ISBN 0-8176-4472-5 , p. 206 .

Individual evidence

1. ^ Wolfram MathWorld, cousin Primes. Retrieved December 1, 2015 .  
2. ↑ 11594 digit cousin prime pair. Retrieved December 1, 2015 .  
3. ↑ Prime pages, . Retrieved December 1, 2015 . ${\displaystyle 474435381\cdot 2^{98394}-1}$ 
4. ↑ B.Segal: generalization you théorème de Brun . Ed .: CR Acad. Sc. URSS. Christine Steyrer, 1930, ISBN 978-3-902662-18-7 , p. 501-507 (Russian).
5. ↑ Zentralblatt MATH Zentralblatt MATH 57.1363.06. Retrieved December 1, 2015 .  
6. ^ Marek Wolf, On the Twin and Cousin Primes ( PostScript file).