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Cousin Primes


Pairs of primes of the form (p, p+4) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).

A large pair of cousin (proven) primes start with

 p={9771919142·[(53238·7879#)^2-1]+2310}·53238·7879#/385+1,
(1)

where 7879# is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).

As of Jan. 2006, the largest known pair of cousin (probable) primes are

 630062·2^(37555)+3,7,
(2)

which have 11311 digits and were found by D. Johnson in May 2004.

According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes,

P_x(p,p+4)∼2product_(p>=3)(p(p-2))/((p-1)^2)int_2^x(dx^')/((lnx^')^2)
(3)
∼1.320323632...int_2^x(dx^')/((lnx^')^2)
(4)

where Pi_2=1.320323632... (OEIS A114907) is the twin primes constant.

An analogy to Brun's constant, the constant

 B_4=(1/7+1/(11))+(1/(13)+1/(17))+(1/(19)+1/(23))+(1/(37)+1/(41))+...,
(5)

(omitting the initial term 1/3+1/7) can be defined. Using cousin primes up to 2^(42), the value of B_4 is estimated as

 B_4 approx 1.1970449.
(6)

See also

Brun's Constant, Prime Constellation, Sexy Primes, Twin Primes, Twin Primes Constant

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References

Andersen, J. K. "Gigantic Sexy and Cousin Primes." Post to primeform user forum. Nov. 3, 2005. http://groups.yahoo.com/group/primeform/message/6637.Sloane, N. J. A. Sequences A023200, A046132, A114907 in "The On-Line Encyclopedia of Integer Sequences."Wolf, M. "On Twin and Cousin Primes." http://www.ift.uni.wroc.pl/~mwolf/.

Referenced on Wolfram|Alpha

Cousin Primes

Cite this as:

Weisstein, Eric W. "Cousin Primes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CousinPrimes.html

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