See Euclidean algorithm on Wiktionary
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Roche, The Mathematics of Measurement: A Critical History, The Athlone Press, page 44:", "text": "The Euclidean algorithms for finding a compound ratio also allowed a ratio and an inverse ratio, and more than two ratios to be compounded, since each compounded pair is equivalent to a single ratio between lines.", "type": "quote" } ], "glosses": [ "Any of certain algorithms first described in Euclid's Elements." ], "id": "en-Euclidean_algorithm-en-noun-usPOj9Ur", "links": [ [ "algorithm", "algorithm" ] ], "raw_glosses": [ "(historical) Any of certain algorithms first described in Euclid's Elements." ], "tags": [ "historical" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Arithmetic", "orig": "en:Arithmetic", "parents": [ "Applied sciences", "Mathematics", "Sciences", "Formal sciences", "All topics", "Fundamental" ], "source": "w" }, { "kind": "topical", "langcode": "en", "name": "Number theory", "orig": "en:Number theory", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "51 49", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "53 47", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "55 45", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" } ], "examples": [ { "text": "1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85, European Conference on Computer Algebra, Linz, Proceedings, Volume 2, Springer, LNCS 204, page 279,\nIn a quadratic field Q (√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. We prove that for D<-19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs." }, { "text": "2003, Ali Akhavi, Brigitte Vallée, Average Bit-Complexity of Euclidean Algorithms, Ugo Montanari, Jose D.P. Rolim, Emo Welzl (editors), Automata, Languages and Programming: 27th International Colloquium, Proceedings, Springer, LNCS 1853, page 373,\nIn this paper, we provide new analyses that characterize the precise average bit-complexity of a class of Euclidean algorithms.\nWe consider here five algorithms that are all classical variations of the Euclidean algorithm and are called Classical (𝒢), By-Excess (ℒ), Centered (𝒦), Subtractive (𝒯) and Binary (ℬ)." }, { "ref": "2009, Brigitte Vallée, Antonio Vera, “3: Probabilistic Analyses of Lattice Reduction Algorithms”, in Phong Q. Nguyen, Brigitte Vallée, editors, The LLL Algorithm: Survey and Applications, Springer, page 71:", "text": "The general behavior of lattice reduction algorithms is far from being well understood.[…]We explain how a mixed methodology has already proved fruitful for small dimensions p, corresponding to the variety of Euclidean algorithms (p = 1) and to the Gauss algorithm (p = 2).", "type": "quote" } ], "glosses": [ "Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method." ], "id": "en-Euclidean_algorithm-en-noun-N03mPVfp", "links": [ [ "arithmetic", "arithmetic" ], [ "number theory", "number theory" ], [ "division", "division" ], [ "algorithm", "algorithm" ], [ "greatest common divisor", "greatest common divisor" ], [ "gcd", "gcd" ], [ "integer", "integer" ] ], "raw_glosses": [ "(arithmetic, number theory) Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method." ], "topics": [ "arithmetic", "mathematics", "number-theory", "sciences" ], "translations": [ { "_dis1": "0 100", "code": "cs", "lang": "Czech", "sense": "number theory", "tags": [ "masculine" ], "word": "Euklidův algoritmus" }, { "_dis1": "0 100", "code": "fi", "lang": "Finnish", "sense": "number theory", "word": "Eukleideen algoritmi" }, { "_dis1": "0 100", "code": "fr", "lang": "French", "sense": "number theory", "tags": [ "masculine" ], "word": "algorithme d’Euclide" }, { "_dis1": "0 100", "code": "de", "lang": "German", "sense": "number theory", "tags": [ "masculine" ], "word": "euklidische Algorithmus" }, { "_dis1": "0 100", "code": "hu", "lang": "Hungarian", "sense": "number theory", "word": "euklideszi algoritmus" }, { "_dis1": "0 100", "code": "is", "lang": "Icelandic", "sense": "number theory", "tags": [ "neuter" ], "word": "reiknirit Evklíðs" }, { "_dis1": "0 100", "code": "it", "lang": "Italian", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritmo di Euclide" }, { "_dis1": "0 100", "code": "ro", "lang": "Romanian", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritm euclidian" }, { "_dis1": "0 100", "code": "ru", "lang": "Russian", "roman": "algorítm Jevklída", "sense": "number theory", "tags": [ "masculine" ], "word": "алгори́тм Евкли́да" }, { "_dis1": "0 100", "code": "es", "lang": "Spanish", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritmo de Euclides" } ] } ], "synonyms": [ { "_dis1": "47 53", "word": "Euclid's algorithm" } ], "wikipedia": [ "Euclidean algorithm" ], "word": "Euclidean algorithm" }
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Roche, The Mathematics of Measurement: A Critical History, The Athlone Press, page 44:", "text": "The Euclidean algorithms for finding a compound ratio also allowed a ratio and an inverse ratio, and more than two ratios to be compounded, since each compounded pair is equivalent to a single ratio between lines.", "type": "quote" } ], "glosses": [ "Any of certain algorithms first described in Euclid's Elements." ], "links": [ [ "algorithm", "algorithm" ] ], "raw_glosses": [ "(historical) Any of certain algorithms first described in Euclid's Elements." ], "tags": [ "historical" ] }, { "categories": [ "English terms with quotations", "en:Arithmetic", "en:Number theory" ], "examples": [ { "text": "1985, Erich Kaltofen, Heinrich Rolletschek, Arithmetic in Quadratic Fields with Unique Factorization, Bob F. Caviness (editor), EUROCAL '85, European Conference on Computer Algebra, Linz, Proceedings, Volume 2, Springer, LNCS 204, page 279,\nIn a quadratic field Q (√), D a squarefree integer, with class number 1 any algebraic integer can be decomposed uniquely into primes but for only 21 domains Euclidean algorithms are known. We prove that for D<-19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs." }, { "text": "2003, Ali Akhavi, Brigitte Vallée, Average Bit-Complexity of Euclidean Algorithms, Ugo Montanari, Jose D.P. Rolim, Emo Welzl (editors), Automata, Languages and Programming: 27th International Colloquium, Proceedings, Springer, LNCS 1853, page 373,\nIn this paper, we provide new analyses that characterize the precise average bit-complexity of a class of Euclidean algorithms.\nWe consider here five algorithms that are all classical variations of the Euclidean algorithm and are called Classical (𝒢), By-Excess (ℒ), Centered (𝒦), Subtractive (𝒯) and Binary (ℬ)." }, { "ref": "2009, Brigitte Vallée, Antonio Vera, “3: Probabilistic Analyses of Lattice Reduction Algorithms”, in Phong Q. Nguyen, Brigitte Vallée, editors, The LLL Algorithm: Survey and Applications, Springer, page 71:", "text": "The general behavior of lattice reduction algorithms is far from being well understood.[…]We explain how a mixed methodology has already proved fruitful for small dimensions p, corresponding to the variety of Euclidean algorithms (p = 1) and to the Gauss algorithm (p = 2).", "type": "quote" } ], "glosses": [ "Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method." ], "links": [ [ "arithmetic", "arithmetic" ], [ "number theory", "number theory" ], [ "division", "division" ], [ "algorithm", "algorithm" ], [ "greatest common divisor", "greatest common divisor" ], [ "gcd", "gcd" ], [ "integer", "integer" ] ], "raw_glosses": [ "(arithmetic, number theory) Specifically, a method, based on a division algorithm, for finding the greatest common divisor (gcd) of two given integers; any of certain variations or generalisations of said method." ], "topics": [ "arithmetic", "mathematics", "number-theory", "sciences" ] } ], "synonyms": [ { "word": "Euclid's algorithm" } ], "translations": [ { "code": "cs", "lang": "Czech", "sense": "number theory", "tags": [ "masculine" ], "word": "Euklidův algoritmus" }, { "code": "fi", "lang": "Finnish", "sense": "number theory", "word": "Eukleideen algoritmi" }, { "code": "fr", "lang": "French", "sense": "number theory", "tags": [ "masculine" ], "word": "algorithme d’Euclide" }, { "code": "de", "lang": "German", "sense": "number theory", "tags": [ "masculine" ], "word": "euklidische Algorithmus" }, { "code": "hu", "lang": "Hungarian", "sense": "number theory", "word": "euklideszi algoritmus" }, { "code": "is", "lang": "Icelandic", "sense": "number theory", "tags": [ "neuter" ], "word": "reiknirit Evklíðs" }, { "code": "it", "lang": "Italian", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritmo di Euclide" }, { "code": "ro", "lang": "Romanian", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritm euclidian" }, { "code": "ru", "lang": "Russian", "roman": "algorítm Jevklída", "sense": "number theory", "tags": [ "masculine" ], "word": "алгори́тм Евкли́да" }, { "code": "es", "lang": "Spanish", "sense": "number theory", "tags": [ "masculine" ], "word": "algoritmo de Euclides" } ], "wikipedia": [ "Euclidean algorithm" ], "word": "Euclidean algorithm" }
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