Editing Twin prime (section)

==Properties==
Usually the pair {{math|(2, 3)}} is not considered to be a pair of twin primes.<ref>
{{cite web
 |title=The first 100,000&nbsp;twin primes (only first member of pair)
 |department=Lists
 |format=plain text
 |website=The Prime Pages (primes.utm.edu)
 |publisher=[[University of Tennessee, Martin|U.T. Martin]]
 |place=Martin, TN
 |url=https://primes.utm.edu/lists/small/100ktwins.txt
}}
</ref>
Since 2 is the only [[parity (mathematics)|even]] prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.

The first several twin prime pairs are
:{{math| (3,  5), (5,  7), (11, 13),}} {{math| (17, 19), (29, 31), (41, 43),}} {{math| (59, 61), (71, 73),  (101, 103),}} {{math| (107, 109), (137, 139), ...}} {{OEIS2C|id=A077800}}.

Five is the only prime that belongs to two pairs, as every twin prime pair greater than {{math| (3, 5) }} is of the form <math>(6n-1, 6n+1)</math> for some [[natural number]] {{mvar|n}}; that is, the number between the two primes is a multiple of 6.<ref>
{{cite web
 |last=Caldwell |first=Chris K.
 |title=Are all primes (past 2 and 3) of the forms {{math|6''n''+1}} and {{math|6''n''−1}}?
 |website=The Prime Pages (primes.utm.edu)
 |publisher=[[University of Tennessee, Martin|U.T. Martin]]
 |place=Martin, TN
 | url=https://primes.utm.edu/notes/faq/six.html
 |access-date=2018-09-27
}}
</ref>
As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.

===Brun's theorem===
{{Main|Brun's theorem}}
In 1915, [[Viggo Brun]] showed that the sum of [[multiplicative inverse|reciprocals]] of the twin primes was [[convergent series|convergent]].<ref name=Brun-1915>
{{cite journal
 |last=Brun |first=V. |author-link=Viggo Brun
 |year=1915
 |title=Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare  |lang=de
 |trans-title=On Goldbach's rule and the number of prime number pairs
 |journal=Archiv for Mathematik og Naturvidenskab
 |volume=34  |number=8  |pages=3–19
 |jfm=45.0330.16  |issn=0365-4524
}}
</ref>
This famous result, called [[Brun's theorem]], was the first use of the [[Brun sieve]] and helped initiate the development of modern [[sieve theory]]. The modern version of Brun's argument can be used to show that the number of twin primes less than {{mvar|N}} does not exceed
:<math>\frac{CN}{(\log N)^2}</math>

for some absolute constant {{nobr|{{mvar|C}} > 0.}}<ref name="Bateman-Diamond-2004">
{{cite book |last1=Bateman |first1=Paul T. |title=Analytic Number Theory |last2=Diamond |first2=Harold G. |publisher=World Scientific |year=2004 |isbn=981-256-080-7 |pages=313 and 334–335 |zbl=1074.11001 |author1-link=Paul T. Bateman}}
</ref>
In fact, it is bounded above by 
<math display=block>\frac{8 C_2 N}{(\log N)^2} \left[ 1 + \operatorname{\mathcal O}\left(\frac{\log \log N}{\log N} \right) \right],</math>
where <math>C_2</math> is the ''twin prime constant'' (slightly less than 2/3), [[#First Hardy–Littlewood conjecture|given below]].<ref>
{{cite book
 |author1=Halberstam, Heini
 |author2=Richert, Hans-Egon
 |year=2010
 |title=Sieve Methods
 |page=117
 |publisher=Dover Publications
}}
</ref>