## Twin Primes |

Twin primes are pairs of primes which differ by two.
The first twin primes are {3,5}, {5,7}, {11,13} and {17,19}.
It has been conjectured
(but never proven) that there are infinitely many twin
primes. *If* the probability of
a random integer *n* and the integer
*n*+2 being prime were statistically independent
events, then it would follow from the
prime number theorem that there are about
*n*/(log *n*)^{2} twin primes less
than or equal to *n*. These probabilities are not
independent, so Hardy and Littlewood
conjectured that the correct estimate should be the following.

Here the infinite product is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...), and we introduce an integral to improve the quality of the estimate. This estimate works quite well! For example:

N | actual | estimate |
---|---|---|

10^{6} | 8169 | 8248 |

10^{8} | 440312 | 440368 |

10^{10} | 27412679 | 27411417 |

There is a longer table by Kutnib and Richstein available online.

In 1919 Brun showed that the sum of the
reciprocals of the twin primes converges to a sum now called
Brun's Constant.
(Recall that the sum of the
reciprocals of all primes diverges.) By calculating the twin primes up to
10^{14} (and discovering the infamous pentium bug along the way),
Thomas Nicely *heuristically*
estimates Brun's constant to be 1.902160578.

As an exercise you might want to prove the following version of Wilson's theorem.

**Theorem:**(Clement 1949)- The integers
*n*,*n*+2, form a pair of twin primes if and only if4[(

*n*-1)!+1] -*n*(mod*n*(*n*+2)).

rank prime digits who when comment 1 3756801695685 · 2^{666669}- 1200700 L1921 Dec 2011 Twin (p) 2 65516468355 · 2^{333333}- 1100355 L923 Aug 2009 Twin (p) 3 4884940623 · 2^{198800}- 159855 L4166 Jul 2015 Twin (p) 4 2003663613 · 2^{195000}- 158711 L202 Jan 2007 Twin (p) 5 38529154785 · 2^{173250}- 152165 L3494 Jul 2014 Twin (p) 6 194772106074315 · 2^{171960}- 151780 x24 Jun 2007 Twin (p) 7 100314512544015 · 2^{171960}- 151780 x24 Jun 2006 Twin (p) 8 16869987339975 · 2^{171960}- 151779 x24 Sep 2005 Twin (p) 9 33218925 · 2^{169690}- 151090 g259 Sep 2002 Twin (p) 10 22835841624 · 7^{54321}- 145917 p296 Nov 2010 Twin (p) 11 1679081223 · 2^{151618}- 145651 L527 Feb 2012 Twin (p) 12 9606632571 · 2^{151515}- 145621 p282 Jul 2014 Twin (p) 13 84966861 · 2^{140219}- 142219 L3121 Apr 2012 Twin (p) 14 12378188145 · 2^{140002}- 142155 L95 Dec 2010 Twin (p) 15 23272426305 · 2^{140001}- 142155 L95 Dec 2010 Twin (p) 16 8151728061 · 2^{125987}- 137936 p35 May 2010 Twin (p) 17 2^{1799}· 3^{137}· 474579581429^{465}· 443749004359^{326}· 644541865141^{488}· 561014826899^{421}· 725590842793^{493}· 623163115793^{476}· 383657519591^{332}- 135851 p360 Dec 2013 Twin (p) 18 598899 · 2^{118987}- 135825 L983 Apr 2010 Twin (p) 19 307259241 · 2^{115599}- 134808 g336 Jan 2009 Twin (p) 20 60194061 · 2^{114689}- 134533 g294 Nov 2002 Twin (p)

- Twin Primes from the World of Mathematics
- The Prime Glossary's: Twin primes

- Forbes97
T. Forbes, "A large pair of twin primes,"Math. Comp.,66(1997) 451-455.MR 97c:11111[The twin primes 6797727 · 2Abstract:We describe an efficient integer squaring algorithm (involving the fast Fourier transform moduloF_{8}) that was used on a 486 computer to discover a large pair of twin primes.^{15328}± 1 are found on a 486 microcomputer]- IJ96
K. IndlekoferandA. Járai, "Largest known twin primes,"Math. Comp.,65(1996) 427-428.MR 96d:11009Abstract:The numbers 697053813 · 2^{16352}± 1 are twin primes.- PSZ90
B. K. Parady,J. F. SmithandS. E. Zarantonello, "Largest known twin primes,"Math. Comp.,55(1990) 381-382.MR 90j:11013

Chris K. Caldwell
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