Editing Prime number (section)

=== Arithmetic progressions ===
{{main|Dirichlet's theorem on arithmetic progressions|Green–Tao theorem}}
An [[arithmetic progression]] is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.<ref>{{cite book |last1=Gelfand |first1=Israel M. |author1-link=Israel Gelfand |url=https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA37 |title=Algebra |last2=Shen |first2=Alexander |publisher=Springer |year=2003 |isbn=978-0-8176-3677-7 |page=37}}</ref> This difference is called the [[Modular arithmetic|modulus]] of the progression.<ref>{{cite book|title=Fundamental Number Theory with Applications|series=Discrete Mathematics and Its Applications|first=Richard A.|last=Mollin|publisher=CRC Press|year=1997|isbn=978-0-8493-3987-5|page=76|url=https://books.google.com/books?id=Fsaa3MUUQYkC&pg=PA76}}</ref> For example,
: <math>3, 12, 21, 30, 39, ...,</math>
is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression
: <math>a, a+q, a+2q, a+3q, \dots</math>
can have more than one prime only when its remainder {{tmath|a}} and modulus {{tmath|q}} are relatively prime. If they are relatively prime, [[Dirichlet's theorem on arithmetic progressions]] asserts that the progression contains infinitely many primes.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA Theorem 1.1.5, p. 12].</ref>
{{Wide image|Prime numbers in arithmetic progression mod&nbsp;9 zoom in.png|815px|Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.|alt=Prime numbers in arithmetic progression mod 9}}
The [[Green–Tao theorem]] shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.<ref name="neale-18-47"/><ref>{{cite journal|first1=Ben|last1=Green|author1-link=Ben J. Green|first2=Terence|last2=Tao|author2-link=Terence Tao|title=The primes contain arbitrarily long arithmetic progressions|journal=[[Annals of Mathematics]]|volume=167|issue=2|year=2008|pages=481–547|doi=10.4007/annals.2008.167.481|arxiv=math.NT/0404188|s2cid=1883951}}</ref>