section_path: Analytic properties/Arithmetic progressions
section_name: Arithmetic progressions
paragraph_text: Arithmetic progressions
  
  An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example,
  
  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
  
  can have more than one prime only when its remainder a {\displaystyle a} and modulus q {\displaystyle q} are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.
  
  The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.