Twin Primes

Twin primes are Primes ($p$, $q$) such that $p-q=2$. The first few twin primes are $n\pm 1$ for $n=4$, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (Sloane's A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (Sloane's A001359 and A006512).

Let $\pi_2(n)$ be the number of twin primes $p$ and $p+2$ such that $p\leq n$. It is not known if there are an infinite number of such Primes (Shanks 1993), but all twin primes except (3, 5) are of the form $6n\pm 1$. J. R. Chen has shown there exists an Infinite number of Primes $p$ such that $p+2$ has at most two factors (Le Lionnais 1983, p. 49). Bruns proved that there exists a computable Integer $x_0$ such that if $x\geq x_0$, then

$$\pi_2(x)< {100x\over(\ln x)^2}$$ (1)

(Ribenboim 1989, p. 201). It has been shown that

$$\pi_2(x)\leq c\prod_{p>2}\left[{1-{1\over(p-1)^2}}\right]{x\... ...left[{1+{\mathcal O}\left({\ln\ln x\over\ln x}\right)}\right],$$ (2)

where $c$ has been reduced to $68/9\approx 7.5556$ (Fouvry and Iwaniec 1983), $128/17\approx 7.5294$ (Fouvry 1984), 7 (Bombieri et al. 1986), 6.9075 (Fouvry and Grupp 1986), and 6.8354 (Wu 1990). The bound on $c$ is further reduced to 6.8325 (Haugland 1999). This calculation involved evaluation of 7-fold integrals and fitting of three different parameters. Hardy and Littlewood conjectured that $c=2$ (Ribenboim 1989, p. 202).

Define

$$E\equiv \liminf_{n\to\infty} {p_{n+1}-p_n\over \ln p_n}.$$ (3)

If there are an infinite number of twin primes, then $E=0$. The best upper limit to date is $E\leq{\textstyle{1\over 4}}+\pi/16=0.44634\ldots$ (Huxley 1973, 1977). The best previous values were 15/16 (Ricci), $(2+\sqrt{3}\,)/8=0.46650\ldots$ (Bombieri and Davenport 1966), and $(2\sqrt{2}-1)/4=0.45706\ldots$ (Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26).

Some large twin primes are $10,006,428\pm 1$, $1,706,595 \times 2^{11235}\pm 1$, and $571,305 \times 2^{7701}\pm 1$. An up-to-date table of known twin primes with 2000 or more digits follows. An extensive list is maintained by Caldwell.

($p, p+1$)	Digits	Reference
$260{,}497{,}545\times 2^{6625}\pm 1$	2003	Atkin and Rickert 1984
$43{,}690{,}485{,}351{,}513\times 10^{1995}\pm 1$	2009	Dubner, Atkin 1985
$2{,}846!!!!\pm 1$	2151	Dubner 1992
$10{,}757{,}0463 \times 10^{2250} \pm 1$	2259	Dubner, Atkin 1985
$663{,}777\times 2^{7650}\pm 1$	2309	Brown et al. 1989
$75{,}188{,}117{,}004\times 10^{2298}\pm 1$	2309	Dubner 1989
$571305\times 2^{7701}\pm 1$	2324	Brown et al. 1989
$1{,}171{,}452{,}282\times 10^{2490}\pm 1$	2500	Dubner 1991
$459\cdot 2^{8529}\pm 1$	2571	Dubner 1993
$1{,}706{,}595\cdot 2^{11235}\pm 1$	3389	Noll et al. 1989
$4{,}655{,}478{,}828\cdot 10^{3429}\pm 1$	3439	Dubner 1993
$1{,}692{,}923{,}232\cdot 10^{4020}\pm 1$	4030	Dubner 1993
$6{,}797{,}727\cdot 2^{15328}\pm 1$	4622	Forbes 1995
$697{,}053{,}813\dot 2^{16352}\pm 1$	4932	Indlekofer and Ja'rai 1994
$570{,}918{,}348\cdot 10^{5120}\pm 1$	5129	Dubner 1995
$242{,}206{,}083\cdot 2^{38880}\pm 1$	11713	Indlekofer and Ja'rai 1995

The last of these is the largest known twin prime pair. In 1995, Nicely discovered a flaw in the Intel ${}^{\scriptstyle\circledRsymbol}$ Pentium${}^{\rm TM}$ microprocessor by computing the reciprocals of 824,633,702,441 and 824,633,702,443, which should have been accurate to 19 decimal places but were incorrect from the tenth decimal place on (Cipra 1995, 1996; Nicely 1996).

If $n\geq 2$, the Integers $n$ and $n+2$ form a pair of twin primes Iff

$$4[(n-1)!+1]+n\equiv 0\ \left({{\rm mod\ } {n(n+2)}}\right).$$ (4)

$n=pp'$ where $(p,p')$ is a pair of twin primes Iff

$$\phi(n)\sigma(n)=(n-3)(n+1)$$ (5)

(Ribenboim 1989). The values of $\pi_2(n)$ were found by Brent (1976) up to $n=10^{11}$. T. Nicely calculated them up to $10^{14}$ in his calculation of Brun's Constant. The following table gives the number less than increasing powers of 10 (Sloane's A007508).

$n$	$\pi_2(n)$
$10^3$	35
$10^4$	205
$10^5$	1224
$10^6$	8,169
$10^7$	58,980
$10^8$	440,312
$10^9$	3,424,506
$10^{10}$	27,412,679
$10^{11}$	224,376,048
$10^{12}$	1,870,585,220
$10^{13}$	15,834,664,872
$10^{14}$	135,780,321,665

See also Brun's Constant, de Polignac's Conjecture Prime Constellation, Sexy Primes, Twin Prime Conjecture, Twin Primes Constant

References

Bombieri, E. and Davenport, H. ``Small Differences Between Prime Numbers.'' Proc. Roy. Soc. Ser. A 293, 1-8, 1966.

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 $10^{11}$.'' Math. Comput. 30, 379, 1976.

Caldwell, C. http://www.utm.edu/cgi-bin/caldwell/primes.cgi/twin.

Cipra, B. ``How Number Theory Got the Best of the Pentium Chip.'' Science 267, 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.

Fouvey, É. and Iwaniec, H. ``Primes in Arithmetic Progression.'' Acta Arith. 42, 197-218, 1983.

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.

Haugland, J. K. Application of Sieve Methods to Prime Numbers. Ph.D. thesis. Oxford, England: Oxford University, 1999.

Huxley, M. N. ``Small Differences between Consecutive Primes.'' Mathematica 20, 229-232, 1973.

Huxley, M. N. ``Small Differences between Consecutive Primes. II.'' Mathematica 24, 142-152, 1977.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Nicely, T. R. ``The Pentium Bug.'' http://lasi.lynchburg.edu/Nicely_T/public/pentbug/pentbug.htm.

Nicely, T. ``Enumeration to $10^{14}$ of the Twin Primes and Brun's Constant.'' Virginia J. Sci. 46, 195-204, 1996. http://lasi.lynchburg.edu/Nicely_T/public/twins/twins.htm.

Parady, B. K.; Smith, J. F.; and Zarantonello, S. E. ``Largest Known Twin Primes.'' Math. Comput. 55, 381-382, 1990.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 199-204, 1989.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 30, 1993.

Sloane, N. J. A. Sequences A014574, A001359/M2476, A006512/M3763, and A007508/M1855 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weintraub, S. ``A Prime Gap of 864.'' J. Recr. Math. 25, 42-43, 1993.

Wu, J. ``Sur la suite des nombres premiers jumeaux.'' Acta. Arith. 55, 365-394, 1990.