Cousin prime

In mathematics, 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.

As of May 2009^[update] the largest known cousin prime was (p, p + 4) for

p = (311778476 · 587502 · 9001# · (587502 · 9001# + 1) + 210)·(587502 · 9001# − 1)/35 + 1

where 9001# is a primorial. It was found by Ken Davis and has 11594 digits.^[2]

The largest known cousin probable prime is

474435381 · 2⁹⁸³⁹⁴ − 1

474435381 · 2⁹⁸³⁹⁴ − 5.

It has 29629 digits and was found by Angel, Jobling and Augustin.^[3] While the first of these numbers has been proven prime, there is no known primality test to easily determine whether the second number is prime.

It follows from the first Hardy–Littlewood conjecture that 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:^[4]

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 B₄ was estimated by Marek Wolf in 1996 as

B₄ ≈ 1.1970449.^[5]

This constant should not be confused with Brun's constant for prime quadruplets, which is also denoted B₄.

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

Notes

^ Weisstein, Eric W. "Cousin Primes". MathWorld.
^ Davis, Ken (2009-05-08). "11594 digit cousin prime pair". primenumbers (Mailing list). Retrieved 2009-05-09. {{cite mailing list}}: Unknown parameter |mailinglist= ignored (|mailing-list= suggested) (help)
^ 474435381 · 2⁹⁸³⁹⁴ − 1. Prime pages.
^ 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 1118045718.
Fine, Benjamin; Rosenberger, Gerhard (2007). Number theory: an introduction via the distribution of primes. Birkhäuser. p. 206. ISBN 0817644725.
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).
Wolf, Marek (February 1998). "Random walk on the prime numbers". Physica A: Statistical Mechanics and its Applications. 250 (1–4): 335–344. doi:10.1016/s0378-4371(97)00661-4.

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

[2] Davis, Ken (2009-05-08). "11594 digit cousin prime pair". primenumbers (Mailing list). Retrieved 2009-05-09. {{cite mailing list}}: Unknown parameter |mailinglist= ignored (|mailing-list= suggested) (help)

[3] 474435381 · 2⁹⁸³⁹⁴ − 1. Prime pages.

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

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

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