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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, 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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