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    This book has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Calude, Cristian S. and Staiger, Ludwig 2018. Liouville, Computable, Borel Normal and Martin-Löf Random Numbers. Theory of Computing Systems, Vol. 62, Issue. 7, p. 1573.


    Vandehey, Joseph 2017. Non-trivial matrix actions preserve normality for continued fractions. Compositio Mathematica, Vol. 153, Issue. 02, p. 274.

    Freeman, David M. 2017. Epicycloid curves and continued fractions. Journal of Mathematics and the Arts, Vol. 11, Issue. 2, p. 100.

    Borwein, Jonathan M. Calkin, Neil J. Lindstrom, Scott B. and Mattingly, Andrew 2017. Continued Logarithms and Associated Continued Fractions. Experimental Mathematics, Vol. 26, Issue. 4, p. 412.

    Baoulina, Ioulia N. and Moree, Pieter 2016. From Arithmetic to Zeta-Functions. p. 1.


Book description

Despite their classical nature, continued fractions are a neverending research area, with a body of results accessible enough to suit a wide audience, from researchers to students and even amateur enthusiasts. Neverending Fractions brings these results together, offering fresh perspectives on a mature subject. Beginning with a standard introduction to continued fractions, the book covers a diverse range of topics, from elementary and metric properties, to quadratic irrationals, to more exotic topics such as folded continued fractions and Somos sequences. Along the way, the authors reveal some amazing applications of the theory to seemingly unrelated problems in number theory. Previously scattered throughout the literature, these applications are brought together in this volume for the first time. A wide variety of exercises guide readers through the material, which will be especially helpful to readers using the book for self-study, and the authors also provide many pointers to the literature.

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[1] B., Adamczewski and Y., Bugeaud, On the Maillet-Baker continued fractions, J. Reine Angew. Math. 606 (2007), 105–121.
[2] W. W., Adams and J. L., Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194–198.
[3] W. W., Adams and M. J., Razar, Multiples of points on elliptic curves and continued fractions, Proc. London Math. Soc. 41 (1980), 481–498.
[4] G., Almkvist and W., Zudilin, Differential equations, mirror maps and zeta values, in: Mirror symmetry y, AMS/IP Stud. Adv. Math. 38 (Amer. Math. Soc., Providence, RI, 2006), pp. 481–515.
[5] G., Andrews and B. C., Berndt, Ramanujan's lost notebook, Parts I, II, II, IV (Springer, New York, 2005, 2009, 2012, 2013).
[6] G. E., Andrews, B. C., Berndt, L., Jacobsen and R. L., Lamphere, The continued fractions found in the unorganized portions of Ramanujan's notebooks, Mem. Amer. Math. Soc. 99 (Amer. Math. Soc., Providence, RI, 1992), no. 477.
[7] F., Apéry, Roger Apéry, 1916-1994: a radical mathematician, Math. Intelligencer 18 (1996), no. 2, 54–61.
[8] R., Apéry, Irrationalité de ζ(2) et ζ(3), Journées arithmétiques de Luminy (20-24 June 1978), Asterisque 61 (Soc. Math.France, Paris, 1979), 11–13.
[9] D. H., Bailey, J. M., Borwein and R. H., Crandall, On the Khintchine constant, Math. Comp. 66 (1997), 417–431.
[10] A., Baker, A concise introduction to the theory of numbers (Cambridge University Press, Cambridge, 1984).
[11] J., Barát and P. P., Varjú, Partitioning the positive integers to seven Beatty sequences, Indag. Math. (NS) 14 (2003), 149–161.
[12] A. F., Beardon and I., Short, The Seidel, Stern, Stolz and Van Vleck theorems on continued fractions, Bull. London Math. Soc. 42 (2010), 457–466.
[13] B. C., Berndt, Ramanujan's notebooks, Parts I, II, III, IV, V (Springer-Verlag, New York, 1985, 1989, 1991, 1994, 1998).
[14] T. G., Berry, On periodicity of continued fractions in hyperelliptic function fields, Arch. Math. (Basel) 55 (1990), 259–266.
[15] M. R., Best and H. J. J., te Riele, On a conjecture of Erdős concerning sums of powers of integers, Report NW 23/76 (Mathematisch Centrum Amsterdam, 1976).
[16] F., Beukers, A note on the irrationality of ζ(2) and ζ(3), Bull. London Math. Soc. 11 (1979), 268–272.
[17] P. E., Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1927), 367–377; Erratum, Math. Ann. 96 (1927), 735.
[18] E., Bombieri and A. J., van der Poorten, Continued fractions of algebraic numbers, in: Computational algebra and number theory, Sydney, 1992, Math. Appl. 325 (Kluwer, Dordrecht, 1995), pp. 137–152.
[19] D., Borwein, J., Borwein, R., Crandall and R., Mayer, On the dynamics of certain recurrence relations, Ramanujan J. 13 (2007), 63–101.
[20] D., Borwein, J. M., Borwein and B., Sims, On the solution of linear mean recurrences, Amer. Math. Monthly (2014), in press.
[21] J., Borwein and D., Bailey, Mathematics by experiment. Plausible reasoning in the 21st century, 2nd edition (A. K. Peters, Wellesley, MA, 2008).
[22] J. M., Borwein, D., Bailey and R., Girgensohn, Experimentation in mathematics: computational paths to discovery (A. K. Peters, Natick, MA, 2004).
[23] J., Borwein and P., Borwein, On the generating function of the integer part: [na + γ], J. Number Theory 43 (1993), no. 3, 293–318.
[24] J., Borwein, P., Borwein and K., Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly 96 (1989), 681–687.
[25] J. M., Borwein, K.-K. S., Choi and W., Pigulla, Continued fractions of tails of hypergeometric series, Amer. Math. Monthly 112 (2005), 493–501.
[26] J., Borwein, R., Crandall and G., Fee, On the Ramanujan AGM fraction. Part I: the real-parameter case, Exp. Math. 13 (2004), 275–286.
[27] J., Borwein and R., Crandall, On the Ramanujan AGM fraction. Part I: the complex-parameter case, Exp. Math. 13 (2004), 287–296.
[28] J. M., Borwein and P., Borwein, Pi and the AGM: a study in analytic number theory and computational complexity (John Wiley, New York, 1987).
[29] J., Borwein and R., Luke, Dynamics of a Ramanujan-type continued fraction with cyclic coefficients, Ramanujan J. 16 (2008), 285–304.
[30] J., Borwein and R., Luke, Dynamics of some random continued fractions, Abstract Appl. Anal. 5 (2005), 449–468.
[31] J. M., Borwein, I., Shparlinski and W., Zudilin (eds.), Number theory and related fields: in memory of Alf van der Poorten, Springer Proc. Math. and Stat. 43 (Springer-Verlag, New York, 2013).
[32] P., Borwein, S., Choi, B., Rooney and A., Weirathmueller, The Riemann hypothesis: a resource for the afficionado and virtuoso alike, CMS Books in Math. (Springer-Verlag, New York, 2007).
[33] J., Bourgain and A., Kontorovich, On Zaremba's conjecture, CR Math. Acad. Sci. Paris Ser. I Math. 349 (2011), 493–495.
[34] D., Bowman, A new generalization of Davison's theorem, Fibonacci Quart. 26 (1988), 40–45.
[35] R. P., Brent, A. J., van der Poorten and H. TE, Riele, A comparative study of algorithms for computing continued fractions of algebraic numbers, in: Algorithmic number theory (Talence, 1996), Lecture Notes in Computer Sci. 1122 (Springer, Berlin, 1996), pp. 35–47.
[36] E. B., Burger, Exploring the number jungle: a journey into Diophantine analysis, Student Math. Library 8 (Amer. Math. Soc., Providence, RI, 2000).
[37] E. B., Burger and T., Struppeck, On frequency distributions of partial quotients of U-numbers, Mathematika 40 (1993), 215–225.
[38] W., Butske, L. M., Jaje and D. R., Mayernik, On the equation, pseudoperfect numbers, and perfectly weighted graphs, Math. Comp. 69 (2000), 407–420.
[39] G., Cairns, N. B., Ho and T., Lengyel, The Sprague-Gundy function of the real game Euclid, Discrete Math. 311 (2011), 457–462.
[40] D. G., Cantor, Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), no. 177, 95–101.
[41] D. G., Cantor, On the analogue of the division polynomials for hyperelliptic curves, J. fur Math. (Crelle) 447 (1994), 91–145.
[42] D. G., Cantor, P. H., Galyean and H. G., Zimmer, A continued fraction algorithm for real algebraic numbers, Math. Comp. 26 (1972), 785–791.
[43] J. W. S., Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Phys. 45 (Cambridge University Press, New York, 1957).
[44] B. M., Char, On Stieltjes' continued fraction for the gamma function, Math. Comp. 34 (1980), 547–551.
[45] S. D., Chowla, Some problems of diophantine approximation (I), Math. Z. 33 (1931), 544–563.
[46] F. W., Clarke, W. N., Everitt, L. L., Littlejohn and S. J. R., Vorster, H. J. S., Smith and the Fermat two squares theorem, Amer. Math. Monthly 106 (1999), 652–665.
[47] H., Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly 113 (2006), 57–62.
[48] R. M., Corless, G. W., Frank and J. G., Monroe, Chaos and continued fractions, Phys. D 46 (1990), 241–253.
[49] T. W., Cusick and M. Mendes, France, The Lagrange spectrum of a set, Acta Arith. 34 (1979), 287–293.
[50] A., Cuyt, V. B., Petersen, B., Verdonk, H., Waadeland and W. B., Jones, Handbook of continued fractions for special functions, with contributions by F., Backeljauw and C., Bonan-Hamada (Springer, New York, 2008).
[51] D. P., Dalzell, On 22/7, J. London Math. Soc. 19 (1944), 133–134.
[52] L. V., Danilov, Some classes of transcendental numbers, Mat. Zametki 12 (1972), 149–154; English translation, Math. Notes Acad. Sci. USSR 12 (1972), 524-527.
[53] H., Davenport, A note on diophantine approximation (II), Mathematika 11 (1964), 50–58.
[54] C. S., Davis, A note on rational approximation, Bull. Austral. Math. Soc. 20 (1979), no. 3, 407–410.
[55] J. L., Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), 29–32.
[56] J. L., Davison and J. O., Shallit, Continued fractions for some alternating series, Monat shefte Math. 111 (1991), 119–126.
[57] P., Erdős, Advanced problem 4347, Amer. Math. Monthly 56 (1949), 343.
[58] S. R., Finch, Mathematical constants, Encyclopedia of Math. and its Applications 94 (Cambridge University Press, Cambridge, 2003).
[59] S., Fomin and A., Zelevinsky, The Laurent phenomenon, Adv. Appl. Math. 28 (2002), 119–144.
[60] L. R., Ford, Fractions, Amer. Math. Monthly 45 (1938), 586–601.
[61] A. S., Fraenkel, The bracket function and complementary sets of integers, Adv. Appl. Math. 28 (2002), 119–144.
[62] A. S., Fraenkel, Complementing and exactly covering sequences, J. Combin. Theory Ser. A 14 (1973), 8–20.
[63] J. S., Frame, Continued fractions and matrices, Amer. Math. Monthly 56 (1949), 98–103.
[64] J., Franel, Les suites de Farey et le problème des nombres premiers, Göttinger Nachrichten (1924), 198–201.
[65] Y., Gallot, P., Moree, and W., Zudilin, The Erdős–Moser equation 1k + 2k + … + (m - 1)k = mk revisited using continued fractions, Math. Comp. 80 (2011), no. 274, 1221–1237.
[66] A. O., Gelfond, Calculus of finite differences, International Monographs on Advanced Math. and Phys. (Hindustan Publishing, Delhi, 1971).
[67] R. L., Graham, Covering the positive integers by disjoint sets of the form {[nα + β]: n = 1, 2,…}, J. Combin. Theory Ser. A 15 (1973), 354–358.
[68] R. L., Graham, D. E., Knuth and O., Patashnik, Concrete mathematics (Addison-Wesley, Reading, MA, 1990).
[69] D. B., Grünberg and P., Moree, Sequences of enumerative geometry: congruences and asymptotics. With an appendix by Don Zagier, Exp. Math. 17 (2008), 409–426.
[70] R. K., Guy, Unsolved problems in number theory, 3rd edition, Problem Books in Math. (Springer, New York, 2004).
[71] D., Hanson, On the product of the primes, Can. Math. Bull. 15 (1972), 33–37.
[72] G. H., Hardy and E. M., Wright, An introduction to the theory of numbers, 5th edition (Oxford University Press, Oxford, 1989).
[73] H. A., Helfgott, Major arcs for Goldbach's theorem, Preprint arXiv: 1305.2897v2 [math. NT] (June 2013).
[74] M., Hirschhorn, Lord Brouncker's continued fraction for π, Math. Gazette 95 (2011), no. 533, 322–326.
[75] A. N. W., Hone, Elliptic curves and quadratic recurrence sequences, Bull. London Math. Soc. 37 (2005), 161–171.
[76] A., Hurwitz and N., Kritikos, Lectures on number theory (Springer-Verlag, Berlin, 1986).
[77] A. E., Ingham, The distribution of prime numbers, Reprint of the 1932 original, with a foreword by R. C., Vaughan, Cambridge Math. Library (Cambridge University Press, Cambridge, 1990).
[78] W. B., Jones and W. J., Thron, Continued fractions: analytic theory and applications, Encyclopedia of Math. and its Applications 11 (Addison-Wesley, Reading, MA, 1980).
[79] B. C., Kellner, Über irreguläre Paare höhere Ordnungen, Diplomarbeit (Math. Inst., Georg-August-Universität zu Göttingen, Germany, 2002); available at http://www.
[80] A., Khintchine, Metrische Kettenbruchprobleme, Compositio Math. 1 (1935), 361–382.
[81] A., Khintchine, Zur metrischen Kettenbruchtheorie, Compositio Math. 3 (1936), 276–285.
[82] A. Ya., Khintchine, Continued fractions, 2nd edition, translated by P., Wynn (P. Noordhoff, Ltd., Groningen, 1963).
[83] D. E., Knuth, The art of computer programming, Vol. II: Seminumerical algorithms (Addison-Wesley, Reading, MA, 1981).
[84] K., Kolden, Continued fractions and linear substitutions, Archiv for Mathematik og Naturvidenskab 50 (1949), 141–196.
[85] T., Komatsu, A certain power series and the inhomogeneous continued fraction expansions, J. Number Theory 59 (1996), 291–312.
[86] T., Komatsu, On inhomogeneous Diophantine approximation with some quasi-periodic expressions, Acta Math. Hungar. 85 (1999), 311–330.
[87] T., Komatsu, On inhomogeneous Diophantine approximation and the Borweins' algorithm, Far East J. Math. Sci. 12 (2004), 203–224.
[88] T., Komatsu, A proof of the continued fraction expansion of e2/s, Integers 7 (2007), no. A30.
[89] C., Krattenthaler, Advanced determinant calculus, in: The Andrews Festschrift (Maratea, 1998), Sém. Lothar. Combin. 42 (1999), Art. B42q, 67 pp.
[90] L., Kuipers and H., Niederreiter, Uniform distribution of sequences (Wiley-Interscience, New York, 1974).
[91] R. O., Kuzmin, On a problem of Gauss, Dokl. Acad. Sci. USSR (1928), 375–380.
[92] J. C., Lagarias and J., Shallit, Linear fractional transformations of continued fractions with bounded partial quotients, J. Théorie Nombres Bordeaux 9 (1997), 267–279; Corrigendum, J. Théorie Nombres Bordeaux 15 (2003), 741-743.
[93] E., Landau, Bemerkungen zu der vorstehenden Abhandlung von Herrn Franel, Göttinger Nachrichten (1924), 198–206.
[94] S., Lang, Introduction to Diophantine approximations, 2nd edition (Springer-Verlag, New York, 1995).
[95] S., Lang and H., Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112–134; Addendum, J. Reine Angew. Math. 267 (1974), 219-220.
[96] D. H., Lehmer, Euclid's algorithm for large numbers, Amer. Math. Monthly 45 (1938), 227–233.
[97] P., Lévy, Sur les lois de probabilité dont dépendent les quotients complets et incomplets d'une fraction continue, Bull. Soc. Math. France 57 (1929), 178–194.
[98] P., Lévy, Sur le développement en fraction continue d'un nombre choisi au hasard, Compositio Math. 3 (1936), 286–303.
[99] P., Liardet and P., Stambul, Algebraic computations with continued fractions, J. Number Theory 73 (1998), 92–121.
[100] F., Lindemann, Über die Zalh π, Math. Ann. 20 (1882), 213–225.
[101] G., Lochs, Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch, Abh. Hamburg Univ. Math. Sem. 27 (1964), 142–144.
[102] L., Lorentzen, Convergence and divergence of the Ramanujan AGM fraction, Ramanujan J. 16 (2008), 83–95.
[103] L., Lorentzen and H., Waadelend, Continued fractions with applications (North Holland, 1992).
[104] J. H., Loxton and A. J., van der Poorten, Arithmetic properties of certain functions in several variables. III, Bull. Austral. Math. Soc. 16 (1977), 15–47.
[105] S. K., Lucas, Approximations to π derived from integrals with nonnegative integrands, Amer. Math. Monthly 116 (2009), 166–172.
[106] K., MacMillan and J., Sondow, Proofs of power sum and binomial coefficient congruences via Pascal's identity, Amer. Math. Monthly 118 (2011), 549–551.
[107] K., MacMillan and J., Sondow, Divisibility of power sums and the generalized Erdős–Moser equation, Elemente Math. 67 (2012), 182–186.
[108] K., Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342–366; Corrigendum, Math. Ann. 103 (1930), 532.
[109] E., Maillet, Introduction à la théorie des nombres transcendants et des propriétés arithmétiques des fonctions (Paris, Gauthier-Villars, 1906).
[110] R., Marcovecchio, The Rhin–Viola method for log 2, Acta Arith. 139 (2009), 147–184.
[111] J., McLaughlin, Symmetry and specializability in the continued fraction expansions of some infinite products, J. Number Theory 127 (2007), 184–219.
[112] N., Möller, On Schönhage's algorithm and subquadratic integer GCD computation, Math. Comp. 77 (2008), 589–607.
[113] P., Moree, Diophantine equations of Erdős–Moser type, Bull. Austral. Math. Soc. 53 (1996), 281–292.
[114] P., Moree, A top hat for Moser's four mathemagical rabbits, Amer. Math. Monthly 118 (2011), 364–370.
[115] P., Moree, H., te Riele and J., Urbanowicz, Divisibility properties of integers x, k satisfying 1k + … + (x - 1)k = xk, Math. Comp. 63 (1994), 799–815.
[116] L., Moser, On the diophantine equation 1n + 2n + 3n + … + (m - 1)n = mn, Scripta Math. 19 (1953), 84–88.
[117] H., Niederreiter, Dyadic fractions with small partial quotients, Monatshefte Math. 101 (1986), 309–315.
[118] Ku., Nishioka, Mahler functions and transcendence, Lecture Notes in Math. 1631 (Springer-Verlag, Berlin, 1996).
[119] Ku., Nishioka, I., Shiokawa and J., Tamura, Arithmetical properties of a certain power series, J. Number Theory 42 (1992), 61–87.
[120] I., Niven, A simple proof that π is irrational, Bull. Amer. Math. Soc. 53 (1947), 509.
[121] I., Niven, Irrational numbers, Carus Math. Monographs 11, Math. Assoc. Amer. (John Wiley, New York, NY, 1956).
[122] K., O'Bryant, A generating function technique for Beatty sequences and other step sequences, J. Number Theory 94 (2002), 299–319.
[123] F. W. J., Olver, D. W., Lozier, R. F., Boisvert and C. W., Clark (eds.), NISThandbook of mathematical functions (Cambridge University Press, New York, 2010).
[124] O., Perron, Uber die Approximation irrationaler Zahlen durch rationale, Sitz. Heidelberg. Akad. Wiss. 12A (1921), 3–17.
[125] O., Perron, Die Lehre von den Kettenbmchen, 3rd edition, Bd. I: Elementare Kettenbrüche (B. G. Teubner, Stuttgart, 1954); Bd. II: Analytisch-funktionentheoretische Kettenbrüche (B. G., Teubner, Stuttgart, 1957).
[126] M., Petkovšek, H. S., Wilf and D., Zeilberger, A = B (A. K. Peters, Wellesley, MA, 1996).
[127] A., van der Poorten, A proof that Euler missed… Apery's proof of the irrationality of ζ(3), Math. Intelligencer 1 (1978/1979), 195–203.
[128] A., van der Poorten, Formal power series and their continued fraction expansion, in: Algorithmic number theory, Lecture Notes in Computer Sci. 1423 (Springer-Verlag, Berlin, 1998), pp. 358–371.
[129] A., van der Poorten, Quadratic irrational integers with partly prescribed continued fraction expansion, Publ. Math. Debrecen 65 (2004), 481–496.
[130] A., van der Poorten, Specialisation and reduction of continued fraction expansions of formal power series, Ramanujan J. 9 (2005), 83–91.
[131] A., van der Poorten, Elliptic curves and continued fractions, J. Integer Sequences 8 (2005), paper 05.2.5, 19 pp.
[132] A., van der Poorten, Curves of genus 2, continued fractions, and Somos sequences, J. Integer Sequences 8 (2005), paper 05.3.4, 9 pp.
[133] A., van der Poorten, Hyperelliptic curves, continued fractions, and Somos sequences, in: Dynamics and stochastics, IMS Lecture Notes Monogr. Ser. 48 (Inst. Math. Statist., Beachwood, OH, 2006), pp. 212-224.
[134] A., van der Poorten and J., Shallit, Folded continued fractions, J. Number Theory 40 (1992), 237-250.
[135] A., van der Poorten and J., Shallit, A specialised continued fraction, Can. J. Math. 45 (1993), 1067-1079.
[136] A. J., van der Poorten and C. S., Swart, Recurrence relations for elliptic sequences: every Somos 4 is a Somos k, Bull. London Math. Soc. 38 (2006), 546-554.
[137] M., Prévost, A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants, J. Comput. Appl. Math. 67 (1996), 219-235.
[138] K., Rajkumar, A simplification of Apéry's proof of the irrationality of ζ(3), Preprint arXiv: 1212.5881 [math. NT] (2012).
[139] G. N., Raney, On continued fractions and finite automata, Math. Ann. 206 (1973), 265-283.
[140] G., Rhin and C., Viola, On a permutation group related to ζ(2), Acta Arith. 77 (1996), 23-56.
[141] R. D., Richtmyer, M., Devaney and N., Metropolis, Continued fraction expansions of algebraic numbers, Numer. Math. 4 (1962), 68-84.
[142] T., Rivoal and W., Zudilin, Diophantine properties of numbers related to Catalan's constant, Math. Ann. 326: 4 (2003), 705-721.
[143] J., Roberts, Elementary number theory: a problem oriented approach (MIT Press, 1978).
[144] A. M., Rockett and P., Szüsz, Continued fractions (World Scientific, Singapore, 1992).
[145] V. Kh., Salikhov, On the irrationality measure of π, Usp. Mat. Nauk. 63 (2008), no. 3, 163-164; English translation, Russian Math. Surveys 63 (2008), 570–572.
[146] A., Schinzel, On some problems of the arithmetical theory of continued fractions, Acta Arith. 6 (1961), 393-413.
[147] A., Schinzel, On some problems of the arithmetical theory of continued fractions II, Acta Arith. 7 (1962), 287-298.
[148] W. M., Schmidt, On badly approximable numbers, Mathematika 12 (1965), 10-20.
[149] W. M., Schmidt, Diophantine approximation, Lecture Notes in Math. 785 (Springer-Verlag, Berlin, 1980).
[150] A., Schönhage, Schnelle Berechnung von Kettenbruchentwicklungen, Acta Informatica 1 (1971), 139-144.
[151] J., Shallit, Real numbers with bounded partial quotients: a survey, L'Enseignement Math. 38 (1992), 151-187.
[152] R., Shipsey, Elliptic divisibility sequences, Ph. D. thesis (Goldsmiths College, University of London, 2000).
[153] P., Shiu, Computation of continued fractions without input values, Math. Comp. 64 (1995), no. 211, 1307-1317.
[154] P., Shiu, A function from Diophantine approximations, Publ. Inst. Math. (Beograd) 65 (1999), 52-62.
[155] Th., Skolem, Über einige Eigenschaften der Zahlenmengen [αn + β] bei irrationalem α mit einleitenden Bemerkungen über einige kombinatorische Probleme, Norske Vid. Selsk. Forh. (Trondheim) 30 (1957), 118-125.
[156] N. J. A., Sloane, The on-line encyclopedia of integer sequences, published electronically at (2013).
[157] K. R., Stromberg, An introduction to classical real analysis (Wadsworth, 1981).
[158] C., Swart, Elliptic curves and related sequences, Ph. D. thesis (Royal Holloway College, University of London, 2003).
[159] B. G., Tasoev, On rational approximations of some numbers, Math. Notes 67 (2000), no. 5–6, 786-791.
[160] R., Tijdeman, Exact covers of balanced sequences and Fraenkel's conjecture, in: Algebraic number theory and Diophantine analysis, Graz, 1998 (de Gruyter, Berlin, 2000), pp. 467-483.
[161] R., Tijdeman, Fraenkel's conjecture for six sequences, Discrete Math. 222 (2000), 223-234.
[162] A. J. H., Vincent, Sur la résolution des équations numériques, J. Math. Pures Appl. 1 (1836), 341-372.
[163] J., Vuillemin, Exact real computer arithmetic with continued fractions, INRIA Report 760 (INRIA, Le Chesnay, France, 1987).
[164] H. S., Wall, Analytic theory of continued fractions (Chelsea Publishing, New York, 1948).
[165] M., Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74.
[166] E. T., Whittaker and G. N., Watson, A course of modern analysis, 4th edition (Cambridge University Press, 1927).
[167] A. J., Yee, γ-cruncher – a multi-threaded pi-program, available at
[168] D. B., Zagier, Zetafunktionen und quadratische Körper (Springer-Verlag, New York–Berlin, 1981).
[169] D. B., Zagier, Problems posed at the St Andrews Colloquium (1996), Solutions, 5th day;
[170] D., Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 (2001), 945-960.
[171] D., Zagier, Integral solutions of Apéry-like recurrence equations, in: Groups and symmetries, CRM Proc. Lecture Notes 47 (Amer. Math. Soc., Providence, RI, 2009), pp. 349-366.
[172] S. K., Zaremba, La méthode des ‘bons treillis’ pour le calcul des intégrales multiples, in: Applications of number theory to numerical analysis, Proc. Sympos., Université de Montréal, 1971 (Academic Press, New York, 1972), pp. 39-119.
[173] Y., Zhang, Bounded gaps between primes, Ann. Math. (2013), in press;
[174] W., Zudilin, Well-poised generation of Apéry-like recursions, J. Comput. Appl. Math. 178 (2005), 513-521.
[175] W., Zudilin, Apéry's theorem. Thirty years after, Intern. J. Math. Computer Sci. 4 (2009), 9-19; An elementary proof of Apéry's theorem, Preprint arXiv:math. NT/0202159 (2002).
[176] W., Zudilin, On the irrationality measure of π2, Usp. Mat. Nauk. 68 (2013), no. 6, 171-172; English translation, Russian Math. Surveys 68 (2013), 1133–1135; Two hypergeometric tales and a new irrationality measure of ζ(2), Preprint arXiv: 1310.1526 [math. NT] (2013).


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