On Euclid’s algorithm and elementary number theory

Hdl Handle:
http://hdl.handle.net/10149/203109
Title:
On Euclid’s algorithm and elementary number theory
Authors:
Backhouse, R. (Roland); Ferreira, J. F. (João)
Affiliation:
Teesside University. School of Computing.
Citation:
Backhouse, R. and Ferreira, J.F. (2011) 'On Euclid’s algorithm and elementary number theory', Science of Computer Programming, 76(3), pp.160-180.
Publisher:
Elsevier
Journal:
Science of Computer Programming
Issue Date:
Mar-2011
URI:
http://hdl.handle.net/10149/203109
DOI:
10.1016/j.scico.2010.05.006
Additional Links:
http://linkinghub.elsevier.com/retrieve/pii/S0167642310000961; http://joaoff.com/publications/2010/euclid-alg/
Abstract:
Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems). The theorems that we verify are well-known and most of them are included in standard number-theory books. The new results concern distributivity properties of the greatest common divisor and a new algorithm for efficiently enumerating the positive rationals in two different ways. One way is known and is due to Moshe Newman. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a Stern-Brocot enumeration algorithm with the same time and space complexity as Newman's algorithm. A short review of the original papers by Stern and Brocot is also included.
Type:
Article
Language:
en
Keywords:
number theory; calculational method; greatest common divisor; Euclid’s algorithm; invariant; Eisenstein array; Eisenstein–Stern tree; Stern–Brocot tree; algorithm derivation; enumeration algorithm; rational number; Calkin–Wilf tree
ISSN:
0167-6423
Rights:
Subject to restrictions, author can archive post-print (ie final draft post-refereeing). For full details see http://www.sherpa.ac.uk/romeo/ [Accessed 16/01/2012]
Citation Count:
1 [Scopus, 16/01/2012]

Full metadata record

DC FieldValue Language
dc.contributor.authorBackhouse, R. (Roland)en
dc.contributor.authorFerreira, J. F. (João)en
dc.date.accessioned2012-01-16T09:21:09Z-
dc.date.available2012-01-16T09:21:09Z-
dc.date.issued2011-03-
dc.identifier.citationScience of Computer Programming; 76(3):160-180en
dc.identifier.issn0167-6423-
dc.identifier.doi10.1016/j.scico.2010.05.006-
dc.identifier.urihttp://hdl.handle.net/10149/203109-
dc.description.abstractAlgorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems). The theorems that we verify are well-known and most of them are included in standard number-theory books. The new results concern distributivity properties of the greatest common divisor and a new algorithm for efficiently enumerating the positive rationals in two different ways. One way is known and is due to Moshe Newman. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a Stern-Brocot enumeration algorithm with the same time and space complexity as Newman's algorithm. A short review of the original papers by Stern and Brocot is also included.en
dc.language.isoenen
dc.publisherElsevieren
dc.relation.urlhttp://linkinghub.elsevier.com/retrieve/pii/S0167642310000961en
dc.relation.urlhttp://joaoff.com/publications/2010/euclid-alg/en
dc.rightsSubject to restrictions, author can archive post-print (ie final draft post-refereeing). For full details see http://www.sherpa.ac.uk/romeo/ [Accessed 16/01/2012]en
dc.subjectnumber theoryen
dc.subjectcalculational methoden
dc.subjectgreatest common divisoren
dc.subjectEuclid’s algorithmen
dc.subjectinvarianten
dc.subjectEisenstein arrayen
dc.subjectEisenstein–Stern treeen
dc.subjectStern–Brocot treeen
dc.subjectalgorithm derivationen
dc.subjectenumeration algorithmen
dc.subjectrational numberen
dc.subjectCalkin–Wilf treeen
dc.titleOn Euclid’s algorithm and elementary number theoryen
dc.typeArticleen
dc.contributor.departmentTeesside University. School of Computing.en
dc.identifier.journalScience of Computer Programmingen
ref.citationcount1 [Scopus, 16/01/2012]en
or.citation.harvardBackhouse, R. and Ferreira, J.F. (2011) 'On Euclid’s algorithm and elementary number theory', Science of Computer Programming, 76(3), pp.160-180.en
All Items in TeesRep are protected by copyright, with all rights reserved, unless otherwise indicated.