New bounds on the length of finite pierce and Engel series
Journal de théorie des nombres de Bordeaux, Volume 3 (1991) no. 1, p. 43-53

Every real number $x,0, has an essentially unique expansion as a Pierce series : $x=\frac{1}{{x}^{1}}-\frac{1}{{x}^{1}{x}^{2}}+\frac{1}{{x}^{1}{x}^{2}{x}^{3}}-\cdots$ where the ${x}_{i}$ form a strictly increasing sequence of positive integers. The expansion terminates if and only if $x$ is rational. Similarly, every positive real number $y$ has a unique expansion as an Engel series : $y=\frac{1}{{y}^{1}}-\frac{1}{{y}^{1}{y}^{2}}+\frac{1}{{y}^{1}{y}^{2}{y}^{3}}+\cdots$ where the ${y}_{i}$ form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi be not eventually constant. Again, such an expansion terminates if and only if $y$ is rational. In this paper we obtain some new upper and lower bounds on the lengths of these series on rational inputs $a/b$. In the case of the Engel series, this answers an open question of Erdös, Rényi, and Szüsz. However, our upper and lower bounds are widely separated.

Classification:  11A67
Keywords: Pierce series, Engel series
@article{JTNB_1991__3_1_43_0,
author = {Erd\"os, P\'al and Shallit, Jeffrey O.},
title = {New bounds on the length of finite pierce and Engel series},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {3},
number = {1},
year = {1991},
pages = {43-53},
zbl = {0727.11003},
mrnumber = {1116100},
language = {en},
url = {http://www.numdam.org/item/JTNB_1991__3_1_43_0}
}

Erdös, P.; Shallit, J. O. New bounds on the length of finite pierce and Engel series. Journal de théorie des nombres de Bordeaux, Volume 3 (1991) no. 1, pp. 43-53. http://www.numdam.org/item/JTNB_1991__3_1_43_0/

1 A. Békéssy, Bemerkungen zur Engleschen Darstellung reeler Zahlen, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 143-151. | MR 102497 | Zbl 0108.26804

2 P. Deheuvels, L'encadrement asymptotique des elements de la série d'Engel d'un nombre réel, C. R. Acad. Sci. Paris 295 (1982), 21-24. | MR 669252 | Zbl 0488.10050

3 F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

4 P. Erdös, A. Rényi, and P. Szüsz, On Engel's and Sylvester's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1 (1958), 7-32. | MR 102496 | Zbl 0107.27002

5 G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1985. | MR 568909

6 M.E. Mays, Iterating the division algorithm, Fibonacci Quart. 25 (1987), 204-213. | MR 901856 | Zbl 0621.10008

7 T.A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly 36 (1929), 523-525. | JFM 55.0305.06 | MR 1521866

8 E. Ya. Remez, On series with alternating signs which may be connected with two algorithms of M. V. Ostrogradskii for the approximation of irrational numbers, Uspekhi Mat. Nauk 6 (5) (1951), 33-42, (MR #13,444d). | MR 44585 | Zbl 0045.02102

9 A. Rényi, A new approach to the theory of Engel's series, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 5 (1962), 25-32. | MR 150123 | Zbl 0232.10028

10 J.B. Rosser and L. Schoenfeld, Approxamate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. | MR 137689 | Zbl 0122.05001

11 J.O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart. 24 (1986), 22-40. | MR 825872 | Zbl 0598.10057

12 J.O. Shallit, Letter to the editor, Fibonacci Quart. 27 (1989), 186.

13 W. Sierpinski, O kilku algorytmach dla rozwijania liczb rzeczywistych na szeregi, C. R. Soc. Sci. Varsovie 4 (1911), 56-77, (In Polish; reprinted in French translation as Sur quelques algorithmes pour développer les nombres reéls en séries, in W. Sierpinski, Oeuvres Choisies, Vol. I, PWN, Warsaw, 1974, pp. 236-254.).

14 K.G. Valeyev and E.D. Zlebov, The metric theory of an algorithm of M. V. Ostrogradskij, Ukrain. Mat. Z. 27 (1975), 64-69. | MR 366855 | Zbl 0309.10020