Euclidean algorithm (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Euclidean algorithm" in English language version.

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Ferguson, H. R. P.; Forcade, R. W. (1979). "Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two". Bulletin of the American Mathematical Society. New Series. 1 (6): 912–914. doi:10.1090/S0273-0979-1979-14691-3. MR 0546316.
Sorenson, Jonathan P. (2004). "An analysis of the generalized binary GCD algorithm". High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications. Vol. 41. Providence, RI: American Mathematical Society. pp. 327–340. ISBN 9780821887592. MR 2076257. The algorithms that are used the most in practice today [for computing greatest common divisors] are probably the binary algorithm and Euclid's algorithm for smaller numbers, and either Lehmer's algorithm or Lebealean's version of the k-ary GCD algorithm for larger numbers.
Bueso, Gómez-Torrecillas & Verschoren (2003); see pp. 37-38 for non-commutative extensions of the Euclidean algorithm and Corollary 4.35, p. 40, for more examples of noncommutative rings to which they apply. Bueso, José; Gómez-Torrecillas, José; Verschoren, Alain (2003). Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups. Mathematical Modelling: Theory and Applications. Vol. 17. Kluwer Academic Publishers, Dordrecht. doi:10.1007/978-94-017-0285-0. ISBN 1-4020-1402-3. MR 2006329.

archive.org

Stark 1978, p. 16 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Stark 1978, p. 21 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Stark 1978, p. 25 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Ore 1948, pp. 47–48 Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill.
Stark 1978, p. 18 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Stark 1978, pp. 16–20 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Ore 1948, p. 43 Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill.
van der Waerden, B. L. (1954). Science Awakening. translated by Arnold Dresden. Groningen: P. Noordhoff Ltd. pp. 114–115.
Heath, T. L. (1949). Mathematics in Aristotle. Oxford Press. pp. 80–83.
Ore 1948, pp. 247–248 Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill.
Lejeune Dirichlet 1894, pp. 29–31 Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie
Richard Dedekind in Lejeune Dirichlet 1894, Supplement XI Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie
Roberts, J. (1977). Elementary Number Theory: A Problem Oriented Approach. Cambridge, MA: MIT Press. pp. 1–8. ISBN 0-262-68028-9.
Stark 1978, pp. 26–36 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Ore 1948, p. 44 Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill.
Stark 1978, pp. 281–292 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Stark 1978, p. 47 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Vinogradov, I. M. (1954). Elements of Number Theory. New York: Dover. pp. 3–13.
Reynaud, A.-A.-L. (1811). Traité d'arithmétique à l'usage des élèves qui se destinent à l'École Polytechnique (6th ed.). Paris: Courcier. Note 60, p. 34. As cited by Shallit (1994).
Ore 1948, p. 45 Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill.
Aho, A.; Hopcroft, J.; Ullman, J. (1974). The Design and Analysis of Computer Algorithms. New York: Addison–Wesley. pp. 300–310. ISBN 0-201-00029-6.
Boyer, C. B.; Merzbach, U. C. (1991). A History of Mathematics (2nd ed.). New York: Wiley. pp. 116–117. ISBN 0-471-54397-7.
Cajori, F (1894). A History of Mathematics. New York: Macmillan. p. 70. ISBN 0-486-43874-0.
Adams, William W.; Goldstein, Larry Joel (1976). Introduction to Number Theory. Prentice-Hall. Exercise 24, p. 205. ISBN 9780134912820. State and prove an analogue of the Chinese remainder theorem for the Gaussian integers.
Stark 1978, p. 290 Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8.
Sharpe, David (1987). Rings and Factorization. Cambridge University Press. p. 55. ISBN 9780521337182.
LeVeque, W. J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57, 81. ISBN 978-0-486-42539-9. Zbl 1009.11001.

arxiv.org

Shor, P. W. (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Journal on Scientific and Statistical Computing. 26 (5): 1484–1509. arXiv:quant-ph/9508027. Bibcode:1995quant.ph..8027S. doi:10.1137/s0097539795293172. S2CID 2337707.

books.google.com

Cohn 1962, p. 104 Cohn, H. (1962). Advanced Number Theory. New York: Dover. ISBN 0-486-64023-X.
Cohen 1993, p. 14 Cohen, H. (1993). A Course in Computational Algebraic Number Theory. New York: Springer-Verlag. ISBN 0-387-55640-0.
Cohen 1993, pp. 14–15, 17–18 Cohen, H. (1993). A Course in Computational Algebraic Number Theory. New York: Springer-Verlag. ISBN 0-387-55640-0.
Sorenson, Jonathan P. (2004). "An analysis of the generalized binary GCD algorithm". High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams. Fields Institute Communications. Vol. 41. Providence, RI: American Mathematical Society. pp. 327–340. ISBN 9780821887592. MR 2076257. The algorithms that are used the most in practice today [for computing greatest common divisors] are probably the binary algorithm and Euclid's algorithm for smaller numbers, and either Lehmer's algorithm or Lebealean's version of the k-ary GCD algorithm for larger numbers.
Joux, Antoine (2009). Algorithmic Cryptanalysis. CRC Press. p. 33. ISBN 9781420070033.
Fuks, D. B.; Tabachnikov, Serge (2007). Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Society. p. 13. ISBN 9780821843161.
Darling, David (2004). "Khintchine's constant". The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes. John Wiley & Sons. p. 175. ISBN 9780471667001.
Williams, Colin P. (2010). Explorations in Quantum Computing. Springer. pp. 277–278. ISBN 9781846288876.
Cox, Little & O'Shea 1997, pp. 37–46 Cox, D.; Little, J.; O'Shea, D. (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (2nd ed.). Springer-Verlag. ISBN 0-387-94680-2.
Grattan-Guinness, Ivor (1990). Convolutions in French Mathematics, 1800-1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics. Volume II: The Turns. Science Networks: Historical Studies. Vol. 3. Basel, Boston, Berlin: Birkhäuser. p. 1148. ISBN 9783764322380. Our subject here is the 'Sturm sequence' of functions defined from a function and its derivative by means of Euclid's algorithm, in order to calculate the number of real roots of a polynomial within a given interval
Hairer, Ernst; Nørsett, Syvert P.; Wanner, Gerhard (1993). "The Routh–Hurwitz Criterion". Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Vol. 8 (2nd ed.). Springer. pp. 81ff. ISBN 9783540566700.
Hensley, Doug (2006). Continued Fractions. World Scientific. p. 26. ISBN 9789812564771.
Dedekind, Richard (1996). Theory of Algebraic Integers. Cambridge Mathematical Library. Cambridge University Press. pp. 22–24. ISBN 9780521565189.
Johnston, Bernard L.; Richman, Fred (1997). Numbers and Symmetry: An Introduction to Algebra. CRC Press. p. 44. ISBN 9780849303012.
Cohn 1962, pp. 104–105 Cohn, H. (1962). Advanced Number Theory. New York: Dover. ISBN 0-486-64023-X.
Lauritzen, Niels (2003). Concrete Abstract Algebra: From Numbers to Gröbner Bases. Cambridge University Press. p. 130. ISBN 9780521534109.
Cohn 1962, pp. 104–110 Cohn, H. (1962). Advanced Number Theory. New York: Dover. ISBN 0-486-64023-X.
Davidoff, Giuliana; Sarnak, Peter; Valette, Alain (2003). "2.6 The Arithmetic of Integer Quaternions". Elementary Number Theory, Group Theory and Ramanujan Graphs. London Mathematical Society Student Texts. Vol. 55. Cambridge University Press. pp. 59–70. ISBN 9780521531436.
Ribenboim, Paulo (2001). Classical Theory of Algebraic Numbers. Universitext. Springer-Verlag. p. 104. ISBN 9780387950709.

doi.org

Shallit, Jeffrey (1994-11-01). "Origins of the analysis of the Euclidean algorithm". Historia Mathematica. 21 (4): 401–419. doi:10.1006/hmat.1994.1031. ISSN 0315-0860.
von Fritz, K. (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics. 46 (2): 242–264. doi:10.2307/1969021. JSTOR 1969021.
Gauss, C. F. (1832). "Theoria residuorum biquadraticorum". Comm. Soc. Reg. Sci. Gött. Rec. 4. Reprinted in Gauss, C. F. (2011). "Theoria residuorum biquadraticorum commentatio prima". Werke. Vol. 2. Cambridge Univ. Press. pp. 65–92. doi:10.1017/CBO9781139058230.004. ISBN 9781139058230. and Gauss, C. F. (2011). "Theoria residuorum biquadraticorum commentatio secunda". Werke. Vol. 2. Cambridge Univ. Press. pp. 93–148. doi:10.1017/CBO9781139058230.005. ISBN 9781139058230.
Ferguson, H. R. P.; Forcade, R. W. (1979). "Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two". Bulletin of the American Mathematical Society. New Series. 1 (6): 912–914. doi:10.1090/S0273-0979-1979-14691-3. MR 0546316.
Cole, A. J.; Davie, A. J. T. (1969). "A game based on the Euclidean algorithm and a winning strategy for it". Math. Gaz. 53 (386): 354–357. doi:10.2307/3612461. JSTOR 3612461. S2CID 125164797.
Spitznagel, E. L. (1973). "Properties of a game based on Euclid's algorithm". Math. Mag. 46 (2): 87–92. doi:10.2307/2689037. JSTOR 2689037.
Shor, P. W. (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Journal on Scientific and Statistical Computing. 26 (5): 1484–1509. arXiv:quant-ph/9508027. Bibcode:1995quant.ph..8027S. doi:10.1137/s0097539795293172. S2CID 2337707.
Dixon, J. D. (1981). "Asymptotically fast factorization of integers". Math. Comput. 36 (153): 255–260. doi:10.2307/2007743. JSTOR 2007743.
Lenstra, H. W. Jr. (1987). "Factoring integers with elliptic curves". Annals of Mathematics. 126 (3): 649–673. doi:10.2307/1971363. hdl:1887/2140. JSTOR 1971363.
Grossman, H. (1924). "On the Number of Divisions in Finding a G.C.D". The American Mathematical Monthly. 31 (9): 443. doi:10.2307/2298146. JSTOR 2298146.
Möller, N. (2008). "On Schönhage's algorithm and subquadratic integer gcd computation" (PDF). Mathematics of Computation. 77 (261): 589–607. Bibcode:2008MaCom..77..589M. doi:10.1090/S0025-5718-07-02017-0.
Tonkov, T. (1974). "On the average length of finite continued fractions". Acta Arithmetica. 26: 47–57. doi:10.4064/aa-26-1-47-57.
Knuth, Donald E. (1976). "Evaluation of Porter's constant". Computers & Mathematics with Applications. 2 (2): 137–139. doi:10.1016/0898-1221(76)90025-0.
Porter, J. W. (1975). "On a Theorem of Heilbronn". Mathematika. 22: 20–28. doi:10.1112/S0025579300004459.
Knuth, D. E. (1976). "Evaluation of Porter's Constant". Computers and Mathematics with Applications. 2 (2): 137–139. doi:10.1016/0898-1221(76)90025-0.
Dixon, J. D. (1970). "The Number of Steps in the Euclidean Algorithm". J. Number Theory. 2 (4): 414–422. Bibcode:1970JNT.....2..414D. doi:10.1016/0022-314X(70)90044-2.
Norton, G. H. (1990). "On the Asymptotic Analysis of the Euclidean Algorithm". Journal of Symbolic Computation. 10: 53–58. doi:10.1016/S0747-7171(08)80036-3.
Stein, J. (1967). "Computational problems associated with Racah algebra". Journal of Computational Physics. 1 (3): 397–405. Bibcode:1967JCoPh...1..397S. doi:10.1016/0021-9991(67)90047-2.
Lehmer, D. H. (1938). "Euclid's Algorithm for Large Numbers". The American Mathematical Monthly. 45 (4): 227–233. doi:10.2307/2302607. JSTOR 2302607.
Sorenson, J. (1994). "Two fast GCD algorithms". J. Algorithms. 16: 110–144. doi:10.1006/jagm.1994.1006.
Weber, K. (1995). "The accelerated GCD algorithm". ACM Trans. Math. Softw. 21: 111–122. doi:10.1145/200979.201042. S2CID 14934919.
Schönhage, A. (1971). "Schnelle Berechnung von Kettenbruchentwicklungen". Acta Informatica (in German). 1 (2): 139–144. doi:10.1007/BF00289520. S2CID 34561609.
Weinberger, P. (1973). "On Euclidean rings of algebraic integers". Proc. Sympos. Pure Math. Proceedings of Symposia in Pure Mathematics. 24: 321–332. doi:10.1090/pspum/024/0337902. ISBN 9780821814246.
Clark, D. A. (1994). "A quadratic field which is Euclidean but not norm-Euclidean". Manuscripta Mathematica. 83: 327–330. doi:10.1007/BF02567617. S2CID 895185. Zbl 0817.11047.
Bueso, Gómez-Torrecillas & Verschoren (2003); see pp. 37-38 for non-commutative extensions of the Euclidean algorithm and Corollary 4.35, p. 40, for more examples of noncommutative rings to which they apply. Bueso, José; Gómez-Torrecillas, José; Verschoren, Alain (2003). Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups. Mathematical Modelling: Theory and Applications. Vol. 17. Kluwer Academic Publishers, Dordrecht. doi:10.1007/978-94-017-0285-0. ISBN 1-4020-1402-3. MR 2006329.

e-rara.ch

Saunderson, Nicholas (1740). The Elements of Algebra in Ten Books. University of Cambridge Press. Retrieved 1 November 2016.

handle.net

hdl.handle.net

Lenstra, H. W. Jr. (1987). "Factoring integers with elliptic curves". Annals of Mathematics. 126 (3): 649–673. doi:10.2307/1971363. hdl:1887/2140. JSTOR 1971363.

harvard.edu

ui.adsabs.harvard.edu

Shor, P. W. (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Journal on Scientific and Statistical Computing. 26 (5): 1484–1509. arXiv:quant-ph/9508027. Bibcode:1995quant.ph..8027S. doi:10.1137/s0097539795293172. S2CID 2337707.
Möller, N. (2008). "On Schönhage's algorithm and subquadratic integer gcd computation" (PDF). Mathematics of Computation. 77 (261): 589–607. Bibcode:2008MaCom..77..589M. doi:10.1090/S0025-5718-07-02017-0.
Dixon, J. D. (1970). "The Number of Steps in the Euclidean Algorithm". J. Number Theory. 2 (4): 414–422. Bibcode:1970JNT.....2..414D. doi:10.1016/0022-314X(70)90044-2.
Stein, J. (1967). "Computational problems associated with Racah algebra". Journal of Computational Physics. 1 (3): 397–405. Bibcode:1967JCoPh...1..397S. doi:10.1016/0021-9991(67)90047-2.

jstor.org

von Fritz, K. (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". Annals of Mathematics. 46 (2): 242–264. doi:10.2307/1969021. JSTOR 1969021.
Cole, A. J.; Davie, A. J. T. (1969). "A game based on the Euclidean algorithm and a winning strategy for it". Math. Gaz. 53 (386): 354–357. doi:10.2307/3612461. JSTOR 3612461. S2CID 125164797.
Spitznagel, E. L. (1973). "Properties of a game based on Euclid's algorithm". Math. Mag. 46 (2): 87–92. doi:10.2307/2689037. JSTOR 2689037.
Dixon, J. D. (1981). "Asymptotically fast factorization of integers". Math. Comput. 36 (153): 255–260. doi:10.2307/2007743. JSTOR 2007743.
Lenstra, H. W. Jr. (1987). "Factoring integers with elliptic curves". Annals of Mathematics. 126 (3): 649–673. doi:10.2307/1971363. hdl:1887/2140. JSTOR 1971363.
Grossman, H. (1924). "On the Number of Divisions in Finding a G.C.D". The American Mathematical Monthly. 31 (9): 443. doi:10.2307/2298146. JSTOR 2298146.
Lehmer, D. H. (1938). "Euclid's Algorithm for Large Numbers". The American Mathematical Monthly. 45 (4): 227–233. doi:10.2307/2302607. JSTOR 2302607.

liu.se

lysator.liu.se

Möller, N. (2008). "On Schönhage's algorithm and subquadratic integer gcd computation" (PDF). Mathematics of Computation. 77 (261): 589–607. Bibcode:2008MaCom..77..589M. doi:10.1090/S0025-5718-07-02017-0.

loc.gov

lccn.loc.gov

Stewart, B. M. (1964). Theory of Numbers (2nd ed.). New York: Macmillan. pp. 43–44. LCCN 64010964.
Lejeune Dirichlet 1894, pp. 29–31 Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie
Richard Dedekind in Lejeune Dirichlet 1894, Supplement XI Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie
Heilbronn, H. A. (1969). "On the Average Length of a Class of Finite Continued Fractions". In Paul Turán (ed.). Number Theory and Analysis. New York: Plenum. pp. 87–96. LCCN 76016027.

sciencenews.org

Peterson, I. (12 August 2002). "Jazzing Up Euclid's Algorithm". ScienceNews.

semanticscholar.org

api.semanticscholar.org

Cole, A. J.; Davie, A. J. T. (1969). "A game based on the Euclidean algorithm and a winning strategy for it". Math. Gaz. 53 (386): 354–357. doi:10.2307/3612461. JSTOR 3612461. S2CID 125164797.
Shor, P. W. (1997). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer". SIAM Journal on Scientific and Statistical Computing. 26 (5): 1484–1509. arXiv:quant-ph/9508027. Bibcode:1995quant.ph..8027S. doi:10.1137/s0097539795293172. S2CID 2337707.
Weber, K. (1995). "The accelerated GCD algorithm". ACM Trans. Math. Softw. 21: 111–122. doi:10.1145/200979.201042. S2CID 14934919.
Schönhage, A. (1971). "Schnelle Berechnung von Kettenbruchentwicklungen". Acta Informatica (in German). 1 (2): 139–144. doi:10.1007/BF00289520. S2CID 34561609.
Clark, D. A. (1994). "A quadratic field which is Euclidean but not norm-Euclidean". Manuscripta Mathematica. 83: 327–330. doi:10.1007/BF02567617. S2CID 895185. Zbl 0817.11047.

siam.org

Cipra, Barry Arthur (16 May 2000). "The Best of the 20th Century: Editors Name Top 10 Algorithms" (PDF). SIAM News. 33 (4). Society for Industrial and Applied Mathematics. Archived from the original (PDF) on 22 September 2016. Retrieved 19 July 2016.

web.archive.org

Cipra, Barry Arthur (16 May 2000). "The Best of the 20th Century: Editors Name Top 10 Algorithms" (PDF). SIAM News. 33 (4). Society for Industrial and Applied Mathematics. Archived from the original (PDF) on 22 September 2016. Retrieved 19 July 2016.

worldcat.org

Shallit, Jeffrey (1994-11-01). "Origins of the analysis of the Euclidean algorithm". Historia Mathematica. 21 (4): 401–419. doi:10.1006/hmat.1994.1031. ISSN 0315-0860.
Lejeune Dirichlet 1894, pp. 29–31 Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie
Richard Dedekind in Lejeune Dirichlet 1894, Supplement XI Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie

zbmath.org

LeVeque, W. J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57, 81. ISBN 978-0-486-42539-9. Zbl 1009.11001.
Clark, D. A. (1994). "A quadratic field which is Euclidean but not norm-Euclidean". Manuscripta Mathematica. 83: 327–330. doi:10.1007/BF02567617. S2CID 895185. Zbl 0817.11047.