Unformatted text preview: queen of mathematics, and the law of quadratic reciprocity the gem of arithmetic. Primes in Arithmetic Progressions • An arithmetic progression is a sequence of the form a,a + b,a + 2 b,a + 3 b,a + 4 b,. .. . • if a and b have a nontrivial common factor, there’s no way this arithmetic progression could contain any primes. • The Theorem: If a and b are relatively prime, then the corresponding arithmetic progression contains in²nitely many primes. • Proved by Dirichlet in 1855. Arithmetic Progressions of Primes • In 2004, Ben Green and Terence Tao proved that there are arbitrarily long (²nite) arithmetic progressions consisting completely of primes....
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- Fall '10
- Number Theory, Arithmetic progression, arithmetic progressions, corresponding arithmetic progression