nt-algorithms

# 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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online