Cousin prime

In number theory, cousin primes are prime numbers that differ by four.^[1] 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 OEIS: A023200 and OEIS: 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 $n, n + 4, n + 8$ will always be divisible by 3, so $n = 3$ is the only case where all three are primes.

An example of a large proven cousin prime pair is $(p, p + 4)$ for

p=4111286921397\times 2^{66420}+1

which has 20008 digits. In fact, this is part of a prime triple since $p$ is also a twin prime (because $p - 2$ is also a proven prime).

As of November 2024^[update], the largest-known pair of cousin primes was found by S. Batalov and has 51,934 digits. The primes are:

Failed to parse (syntax error): {\displaystyle p = (42550837315 \times (2^{110503} - 1) + 1) \times 2^{110505} - 3}

Failed to parse (syntax error): {\displaystyle p+4 = (42550837315 \times (2^{110503} - 1) + 1) \times 2^{110505} + 1} ^[2]

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:^[3]

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 .

Using cousin primes up to 2⁴², the value of $B 4$ was estimated by Marek Wolf in 1996 as

B_{4}\approx 1.1970449

^[4]

This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted $B 4$ .

The Skewes number for cousin primes is 5206837 (Tóth (2019)).

Notes

^ Weisstein, Eric W. "Cousin Primes". MathWorld.
^ Batalov, S. "Let's find some large sexy prime pair[s]". mersenneforum.org. Retrieved 2022-09-17.
^ Segal, B. (1930). "Generalisation du théorème de Brun". C. R. Acad. Sci. URSS (in Russian). 1930: 501–507. JFM 57.1363.06.
^ Marek Wolf (1996), On the Twin and Cousin Primes.

References

Wells, David (2011). Prime Numbers: The Most Mysterious Figures in Math. John Wiley & Sons. p. 33. ISBN 978-1118045718.
Fine, Benjamin; Rosenberger, Gerhard (2007). Number theory: an introduction via the distribution of primes. Birkhäuser. pp. 206. ISBN 978-0817644727.
Tóth, László (2019), "On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood" (PDF), Computational Methods in Science and Technology, 25 (3), arXiv:1910.02636, doi:10.12921/cmst.2019.0000033.
Wolf, Marek (February 1998). "Random walk on the prime numbers". Physica A: Statistical Mechanics and Its Applications. 250 (1–4): 335–344. Bibcode:1998PhyA..250..335W. doi:10.1016/s0378-4371(97)00661-4.

[1] Weisstein, Eric W. "Cousin Primes". MathWorld.

[2] Batalov, S. "Let's find some large sexy prime pair[s]". mersenneforum.org. Retrieved 2022-09-17.

[3] Segal, B. (1930). "Generalisation du théorème de Brun". C. R. Acad. Sci. URSS (in Russian). 1930: 501–507. JFM 57.1363.06.

[4] Marek Wolf (1996), On the Twin and Cousin Primes.

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