Innumber theory,cousin primes areprime numbers that differ by four.^[1] Compare this withtwin primes, pairs of prime numbers that differ by two, andsexy primes, pairs of prime numbers that differ by six.

The cousin primes (sequencesOEIS: A023200 andOEIS: A046132 inOEIS) 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)

The only prime belonging to two pairs of cousin primes is 7. One of the numbersn,n + 4,n + 8 will always be divisible by 3, son = 3 is the only case where all three are primes.

An example of a largeproven cousin prime pair is(p,p + 4) for

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

which has 20008 digits. In fact, this is part of aprime triple sincep is also atwin prime (becausep – 2 is also a proven prime).

As of December 2024^[update], the largest-known pair of cousin primes was found by S. Batalov and has 86,138 digits. The primes are:

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

${\displaystyle p+4=(29571282950\times (2^{190738}-1)+4)\times 2^{95369}+3}$^[2]

If the firstHardy–Littlewood conjecture holds, then cousin primes have the same asymptotic density astwin primes. An analogue ofBrun's constant for twin primes can be defined for cousin primes, calledBrun's constant for cousin primes, with the initial term (3, 7) omitted, by the convergent sum:^[3]

${\displaystyle B_4=\left(\frac{1}{7}+\frac{1}{11}\right)+\left(\frac{1}{13}+\frac{1}{17}\right)+\left(\frac{1}{19}+\frac{1}{23}\right)+\cdots .}$

Using cousin primes up to 2⁴², the value ofB₄ was estimated by Marek Wolf in 1996 as

${\displaystyle B_4\approx 1.1970449}$^[4]

This constant should not be confused with Brun's constant forprime quadruplets, which is also denotedB₄.

TheSkewes number for cousin primes is 5206837 (Tóth (2019)).

• 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.

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