Cousin prime

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

:$p\; =\; 4111286921397\; \backslash times\; 2^\{66420\}\; +\; 1$

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:

:$p\; =\; (29571282950\; \backslash times\; (2^\{190738\}\; -\; 1)\; +\; 4)\; \backslash times\; 2^\{95369\}\; -\; 1$

:$p+4\; =\; (29571282950\; \backslash times\; (2^\{190738\}\; -\; 1)\; +\; 4)\; \backslash times\; 2^\{95369\}\; +\; 3${{cite web |first=S. |last=Batalov |url=https://www.mersenneforum.org/showthread.php?p=605196#post605196 |title=Let's find some large sexy prime pair[s] |website=mersenneforum.org |access-date=2022-09-17}}

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 }}

:$B\_4\; =\; \backslash left(\backslash frac\{1\}\{7\}\; +\; \backslash frac\{1\}\{11\}\backslash right)\; +\; \backslash left(\backslash frac\{1\}\{13\}\; +\; \backslash frac\{1\}\{17\}\backslash right)\; +\; \backslash left(\backslash frac\{1\}\{19\}\; +\; \backslash frac\{1\}\{23\}\backslash right)\; +\; \backslash cdots.$

Using cousin primes up to 2⁴², the value of {{math|B₄}} was estimated by Marek Wolf in 1996 as

:$B\_4\; \backslash approx\; 1.1970449$Marek Wolf (1996), [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.1864 On the Twin and Cousin Primes].

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}}).

{{reflist}}

• {{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}}

Category:Classes of prime numbers

Category:Unsolved problems in mathematics