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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, theres no way this arithmetic progression could contain any primes. The Theorem: If a and b are relatively prime, then the corresponding arithmetic progression contains innitely many primes. Proved by Dirichlet in 1855. Arithmetic Progressions of Primes In 2004, Ben Green and Terence Tao proved that there are arbitrarily long (nite) arithmetic progressions consisting completely of primes....
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- Fall '10