Bombieri, E.; Friedlander, J. B.; and Iwaniec, H. "Primes in Arithmetic Progression to Large Moduli." Acta Math.156, 203-251, 1986.Bradley, C. J. "The Location of Twin Primes." Math. Gaz.67, 292-294, 1983.Brent, R. P. "Irregularities in the Distribution of Primes and Twin Primes." Math. Comput.29, 43-56, 1975.Brent, R. P. "UMT 4." Math. Comput.29, 221, 1975.Brent, R. P. "Tables Concerning Irregularities in the Distribution of Primes and Twin Primes to ." Math. Comput.30, 379, 1976.Caldwell, C. http://primes.utm.edu/top20/page.php?id=1.Caldwell, C. K. "The Top Twenty: Twin Primes." http://www.utm.edu/research/primes/lists/top20/twin.html.Cipra, B. "How Number Theory Got the Best of the Pentium Chip." Science267, 175, 1995.Cipra, B. "Divide and Conquer." What's Happening in the Mathematical Sciences, 1995-1996, Vol. 3. Providence, RI: Amer. Math. Soc., pp. 38-47, 1996.Fouvry, É. "Autour du théorème de Bombieri-Vinogradov." Acta Math.152, 219-244, 1984.Fouvry, É. and Grupp, F. "On the Switching Principle in Sieve Theory." J. reine angew. Math.370, 101-126, 1986.Fouvry, É. and Iwaniec, H. "Primes in Arithmetic Progressions." Acta Arith.42, 197-218, 1983.Fry, P.; Nesheiwat, J.; and Szymanski, B. K. "Experiences with Distributed Computation of Twin Primes Distribution." In Progress in Computer Research, Vol. 2. (Ed. F. Columbus). Commack, NY: Nova Science Pub., pp. 187-203, 2001.Gardner, M. "Patterns in Primes are a Clue to the Strong Law of Small Numbers." Sci. Amer.243, 18-28, Dec. 1980.Gourdon, X. and Sebah, P. "Introduction to Twin Primes and Brun's Constant Computation." http://numbers.computation.free.fr/Constants/Primes/twin.html.Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 19-23, 1994.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Haugland, J. K. Application of Sieve Methods to Prime Numbers. Ph.D. thesis. Oxford, England: Oxford University, 1999.Indlekofer, K. H. and Járai, A. "Largest Known Twin Primes." Math. Comput.65, 427-428, 1996.Indlekofer, K. H. and Járai, A. "Largest Known Twin Primes and Sophie Germain Primes." Math. Comput.68, 1317-1324, 1999.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1979.Nicely, T. R. "The Pentium Bug." http://www.trnicely.net/pentbug/pentbug.html.Nicely, T. R. "Enumeration to of the Twin Primes and Brun's Constant." Virginia J. Sci.46, 195-204, 1996. http://www.trnicely.net/twins/twins.html.Nicely, T. R. "New Maximal Prime Gaps and First Occurrences." Math. Comput.68, 1311-1315, 1999.Nyman, B. and Nicely, T. R. "New Prime Gaps Between and ." J. Int. Seq.6, 1-6, 2003.Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. "Largest Known Twin Primes." Math. Comput.55, 381-382, 1990.Ribenboim, P. "Twin Primes." §4.3 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 259-265, 1996.Sebah, P. "Counting Twin Primes and Brun's Constant New Computation" 22 Aug 2002. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0208&L=nmbrthry&P=1968.Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993.Sloane, N. J. A. Sequences A001359/M2476, A006512/M3763, A007508/M1855, A007534, and A014574 in "The On-Line Encyclopedia of Integer Sequences."Tietze, H. "Prime Numbers and Prime Twins." Ch. 1 in Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, pp. 1-20, 1965.Weintraub, S. "A Prime Gap of 864." J. Recr. Math.25, 42-43, 1993.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 41, 1986.Wolf, M. "On Twin and Cousin Primes." http://www.ift.uni.wroc.pl/~mwolf/.Wolf, M. "Some Conjectures on the Gaps Between Consecutive Primes." http://www.ift.uni.wroc.pl/~mwolf/.Wu, J. "Sur la suite des nombres premiers jumeaux." Acta. Arith.55, 365-394, 1990.
, 
) 的对。术语“孪生素数”由 Paul Stäckel (1862-1919; Tietze 1965, p. 19) 创造。前几个孪生素数是 
对于 
, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574)。明确地，这些是 (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 和 A006512)。
。
。
, 
, 
)、表兄弟素数 (
, 
)、性感素数 (
, 
) 等的 
, 
)
, 
)
, 
)
, 
)
, 
)
, 
)
为孪生素数 
和 
的数量，使得 
。素数 (Wells 1986, p. 41; Shanks 1993) 是否有无限个这样的素数尚不清楚，但似乎几乎可以肯定是真的 (Hardy and Wright 1979, p. 5)。
，使得 
最多有两个因子 (Le Lionnais 1979, p. 49)。Brun 证明存在一个可计算的 整数 
，使得如果 
，则
![pi_2(x)<=cproduct_(p>2)[1-1/((p-1)^2)]x/((lnx)^2)[1+O((lnlnx)/(lnx))],](/images/equations/TwinPrimes/NumberedEquation3.svg)
![pi_2(x)<=cPi_2x/((lnx)^2)[1+O((lnlnx)/(lnx))],](/images/equations/TwinPrimes/NumberedEquation4.svg)
被称为 孪生素数常数，
是另一个常数。常数 
已被简化为 
(Fouvry and Iwaniec 1983), 
(Fouvry 1984), 7 (Bombieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), 6.8354 (Wu 1990) 和 6.8325 (Haugland 1999)。后一个计算涉及 7 重积分的评估和三个不同参数的拟合。
(Ribenboim 1996, p. 262)，并且 
渐近地等于
是第 
个素数，
是 素数差函数，满足 
。![pi_2(x)∼2Pi_2([pi(x)]^2)/x,](/images/equations/TwinPrimes/NumberedEquation7.svg)
）比 
更符合数值数据，但不如 
。
，使得 
改变符号。Wolf 检查了高达 
的数字，发现超过 
个符号变化。根据这些数据，Wolf 推测 
的符号变化数 
的 
由下式给出
位十进制数字，由 PrimeGrid 于 2011 年 12 月 25 日发现 (http://primes.utm.edu/top20/page.php?id=1#records)。
和 
的倒数发现 Intel® PentiumTM 微处理器中的一个缺陷，这些倒数应该精确到小数点后 19 位，但从小数点后第十位开始就不正确了 (Cipra 1995, 1996; Nicely 1996)。
，则 
和 
形成一对孪生素数 ![4[(n-1)!+1]+n=0 (mod n(n+2)).](/images/equations/TwinPrimes/NumberedEquation10.svg)
其中 
是一对孪生素数 当且仅当
是孪生素数 当且仅当
，其中 
是 向下取整函数。
的值由 Brent (1976) 发现，最高达 
。T. Nicely 在计算 布朗常数 时将它们计算到 
。Fry et al. (2001) 和 Sebah (2002) 独立获得了 
，使用了分布式计算。下表给出了 
的已知值 (OEIS A007508; Ribenboim 1996, p. 263; Nicely 1999; Sebah 2002)。
." Math. Comput. 30, 379, 1976.Caldwell, C. 
of the Twin Primes and Brun's Constant." Virginia J. Sci. 46, 195-204, 1996. 
and 
." J. Int. Seq. 6, 1-6, 2003.Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. "Largest Known Twin Primes." Math. Comput. 55, 381-382, 1990.Ribenboim, P. "Twin Primes." §4.3 in