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Modular Arithmetic Formalized and systematized by Gauss in Disquisitiones arithmeticae (1801). m and n are congruent modulo p ( m n ( mod p) ) if p is a divisor of the di±erence of m and n . Clock arithmetic. How is it similar to or di±erent from ordinary arithmetic? How can it be used to solve Diophantine equations like x 2 5 y 2 = 2? Chinese Remainder Theorem The Theorem: If p 1 ,p 2 ,...,p k are relatively prime and a 1 ,a 2 ,...,a k are given numbers. There is a number x such that x a i ( mod p i ) for each i (and there is a practical way to compute such an x ). Practicality Example Quadratic Reciprocity m is a quadratic residue mod p if there is a number x such that m x 2 ( mod p) . The Legendre symbol: p p q P = b 1 if p is a quadratic residue modulo q 1 otherwise The Theorem: If p and q are odd primes then p p q Pp q p P = ( 1 ) (p 1 )(q 1 )/ 4 . Conjectured by Euler. Proved by Gauss. Gauss called mathematics the queen of the sciences, arithmetic the
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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, 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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