Cousin prime
{{Short description|Prime numbers which differ by 4}}
In number theory, cousin primes are prime numbers that differ by four.{{MathWorld|urlname=CousinPrimes|title=Cousin Primes}} Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six.
The cousin primes (sequences {{OEIS2C|id=A023200}} and {{OEIS2C|id=A046132}} in OEIS) below 1000 are:
:(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)
Properties
The only prime belonging to two pairs of cousin primes is 7. One of the numbers {{math|n, n + 4, n + 8}} will always be divisible by 3, so {{math|1=n = 3}} is the only case where all three are primes.
An example of a large proven cousin prime pair is {{math|(p, p + 4)}} for
:
which has 20008 digits. In fact, this is part of a prime triple since {{mvar|p}} is also a twin prime (because {{math|p − 2}} is also a proven prime).
{{As of|2024|December}}, the largest-known pair of cousin primes was found by S. Batalov and has 86,138 digits. The primes are:
:
If the first Hardy–Littlewood conjecture holds, then cousin primes have the same asymptotic density as twin primes. An analogue of Brun's constant for twin primes can be defined for cousin primes, called Brun's constant for cousin primes, with the initial term (3, 7) omitted, by the convergent sum:{{cite journal | last=Segal | first=B. | title=Generalisation du théorème de Brun | language=Russian | jfm=57.1363.06 | journal=C. R. Acad. Sci. URSS | volume=1930 | pages=501–507 | year=1930 }}
:
Using cousin primes up to 242, the value of {{math|B4}} was estimated by Marek Wolf in 1996 as
This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted {{math|B4}}.
The Skewes number for cousin primes is 5206837 ({{harvtxt|Tóth|2019}}).
Notes
{{reflist}}
References
- {{cite book | title=Prime Numbers: The Most Mysterious Figures in Math | first=David | last=Wells | publisher=John Wiley & Sons | year=2011 | isbn=978-1118045718 | page=33 }}
- {{cite book | title=Number theory: an introduction via the distribution of primes | url=https://archive.org/details/numbertheoryintr00fine_621 | url-access=limited | first1=Benjamin | last1=Fine | first2=Gerhard | last2=Rosenberger | publisher=Birkhäuser | year=2007 | isbn=978-0817644727 | pages=[https://archive.org/details/numbertheoryintr00fine_621/page/n214 206] }}
- {{citation|first=László|last=Tóth|title=On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood|journal=Computational Methods in Science and Technology|volume=25|year=2019|issue=3|url=http://cmst.eu/wp-content/uploads/files/10.12921_cmst.2019.0000033_TOTH.pdf|doi=10.12921/cmst.2019.0000033|arxiv=1910.02636|doi-access=free}}.
- {{cite journal|last1=Wolf|first1=Marek|title=Random walk on the prime numbers|journal=Physica A: Statistical Mechanics and Its Applications|date=February 1998|volume=250|issue=1–4|pages=335–344|doi=10.1016/s0378-4371(97)00661-4|bibcode=1998PhyA..250..335W}}
{{Prime number classes}}