Twin Primes

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The Prime Pages

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page.This page is about one of those forms.

Definitions and Notes

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:

The number of twin primes
less than N
N	actual	estimate
10⁶	8169	8248
10⁸	440312	440368
10¹⁰	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 10¹⁴ (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 if
4[(n-1)!+1] ≡ -n (mod n(n+2)).

Nice--too bad it is of virtually no practical value!

Record Primes of this Type

rank prime digits who when comment

1 2996863034895 · 2^1290000 - 1 388342 L2035 Sep 2016 Twin (p)
2 3756801695685 · 2⁶⁶⁶⁶⁶⁹ - 1 200700 L1921 Dec 2011 Twin (p)
3 65516468355 · 2³³³³³³ - 1 100355 L923 Aug 2009 Twin (p)
4 160204065 · 2²⁶²¹⁴⁸ - 1 78923 L5115 Jul 2021 Twin (p)
5 12770275971 · 2²²²²²⁵ - 1 66907 L527 Jul 2017 Twin (p)
6 12599682117 · 2²¹¹⁰⁸⁸ - 1 63554 L4166 Feb 2022 Twin (p)
7 12566577633 · 2²¹¹⁰⁸⁸ - 1 63554 L4166 Feb 2022 Twin (p)
8 70965694293 · 2²⁰⁰⁰⁰⁶ - 1 60219 L95 Apr 2016 Twin (p)
9 66444866235 · 2²⁰⁰⁰⁰³ - 1 60218 L95 Apr 2016 Twin (p)
10 4884940623 · 2¹⁹⁸⁸⁰⁰ - 1 59855 L4166 Jul 2015 Twin (p)
11 2003663613 · 2¹⁹⁵⁰⁰⁰ - 1 58711 L202 Jan 2007 Twin (p)
12 17976255129 · 2¹⁸³²⁴¹ - 1 55172 p415 May 2021 Twin (p)
13 191547657 · 2¹⁷³³⁷² - 1 52199 L5116 Jul 2020 Twin (p)
14 38529154785 · 2¹⁷³²⁵⁰ - 1 52165 L3494 Jul 2014 Twin (p)
15 194772106074315 · 2¹⁷¹⁹⁶⁰ - 1 51780 x24 Jun 2007 Twin (p)
16 100314512544015 · 2¹⁷¹⁹⁶⁰ - 1 51780 x24 Jun 2006 Twin (p)
17 16869987339975 · 2¹⁷¹⁹⁶⁰ - 1 51779 x24 Sep 2005 Twin (p)
18 33218925 · 2¹⁶⁹⁶⁹⁰ - 1 51090 g259 Sep 2002 Twin (p)
19 110427610 · 3¹⁰⁰⁰⁰³ - 1 47722 p415 Feb 2021 Twin (p)
20 3706785456 · 13⁴²⁰⁶⁹ - 1 46873 p412 Sep 2020 Twin (p)

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

References

Forbes97
T. Forbes, "A large pair of twin primes," Math. Comp., 66 (1997) 451-455. MR 97c:11111
Abstract: We describe an efficient integer squaring algorithm (involving the fast Fourier transform modulo F₈) that was used on a 486 computer to discover a large pair of twin primes.
[The twin primes 6797727 · 2¹⁵³²⁸± 1 are found on a 486 microcomputer]
IJ96
K. Indlekofer and A. Járai, "Largest known twin primes," Math. Comp., 65 (1996) 427-428. MR 96d:11009
Abstract: The numbers 697053813 · 2¹⁶³⁵²± 1 are twin primes.

PSZ90
B. K. Parady, J. F. Smith and S. E. Zarantonello, "Largest known twin primes," Math. Comp., 55 (1990) 381-382. MR 90j:11013