Nt-algorithms - Algorithms in Number Theory Jonathan L.F King University of Florida Gainesville FL 32611-2082 USA 24 April 2011(at 01:16 Proof From

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Algorithms in Number Theory Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA 24 April, 2011 (at 01:16 ) Iterated Lightning-bolt ( Euclidean algorithm ) Fix integers J 0 and J 1 , and set D := Gcd( J 0 ,J 1 ). A pair ( s,t ) of integers is a ezout pair for J 0 ,J 1 if sJ 0 + tJ 1 = D . 1a : ezout’s lemma says: There always exists a B´ ezout pair. ( Alternative term: s and t are ezout multipliers . ) A B´ ezout pair ( s,t ) is not unique; it is ( except in the boring J 0 =0= J 1 case ) part of a one-parameter family s k := s + h k · J 1 D i and t k := t - h k · J 0 D i , 1b : of B´ ezout pairs ( s k ,t k ), for each k Z . 1c : Exercise. Prove that (1b) describes all the B´ ezout pairs for J 0 ,J 1 . ± Gcd of several integers. Given a list of integers, ~ J = ( J 0 ,J 1 ,...,J L ), use Gcd( J 0 ,J 1 ,...,J L ) or Gcd( ~ J ) 2a : to denote the greatest common divisor, D , of the list. Our goal is to simultaneously compute D and B´ ezout multipliers ~ s := ( s 0 ,...,s L ) such that X L =0 [ s · J ] = D . 2b : We’ll accomplish this with L applications of LBolt : D note ==== Gcd ± ... Gcd ( Gcd( J 0 ,J 1 ) ,J 2 ) ...,J L ² . Algorithm: From integers ~ J = ( J 0 ,J 1 ,...,J L - 1 ,J L ), set C := Gcd( J 0 ,J 1 ,...,J L - 1 ) and D := Gcd( J 0 ,J 1 ,...,J L - 1 ,J L ) note ==== Gcd( C,J L ) . Apply
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This note was uploaded on 01/26/2012 for the course MHF 3202 taught by Professor Larson during the Fall '09 term at University of Florida.

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Nt-algorithms - Algorithms in Number Theory Jonathan L.F King University of Florida Gainesville FL 32611-2082 USA 24 April 2011(at 01:16 Proof From

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