generate questions: can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.
What is another way to state the condition that infinitely many primes can exist only if a and q are coprime? {sep_token} If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of primes? {sep_token} What is the density of all primes compatible with a modulo 9? {sep_token} If q=9 and a=3,6 or 9, how many primes would be in the progression? {sep_token} If q=9 and a=1,2,4,5,7, or 8, how many primes would be in a progression?