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Transactions of the American Mathematical Society

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Sums of $k$ unit fractions

Author: Christian Elsholtz
Journal: Trans. Amer. Math. Soc. 353 (2001), 3209-3227
MSC (2000): Primary 11D68; Secondary 11D72, 11N36
Published electronically: April 12, 2001
MathSciNet review: 1828604
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Abstract | References | Similar Articles | Additional Information


Erdos and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac{4}{n}= \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$has a solution in positive integers $x,\, y$, and $z$. Let $m > k \geq 3$ and

\begin{displaymath}E_{m,k}(N)= \, \mid \{ n \leq N \mid \frac{m}{n} = \frac{1}... ...t_k} \text{ has no solution with }t_i \in \mathbb{N}\} \mid . \end{displaymath}

We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$where the number of parameters increases exponentially with $k$. This enables us to prove

\begin{displaymath}E_{m,k}(N) \ll N \exp \left( -c_{m,k} (\log N)^{1-\frac{1}{2^{k-1}-1}} \right) \text{ with } c_{m,k}>0. \end{displaymath}

This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization'' of the work of Vaughan (1970), who considered the case $k=3$.

References [Enhancements On Off] (What's this?)

  • [AB98] M.H. Ahmadi and M.N. Bleicher.
    On the conjectures of Erdos and Straus, and Sierpinski on Egyptian fractions.
    Int. J. Math. Stat. Sci., 7:169-185, 1998.
    See also Zentralblatt 990.17875. MR 99k:11049
  • [Cro00] E.S. Croot.
    Unit Fractions.
    PhD thesis, University of Georgia, Athens, 2000.
    The thesis is based on three papers: 1) On some questions of Erdos and Graham about Egyptian fractions, to appear in Mathematika, 2) On unit fractions with denominators in short intervals, to appear in Acta Arithmetica, 3) On a coloring conjecture about unit fractions.
  • [Ded31] R. Dedekind.
    Über Zerlegungen von Zahlen durch ihren größten gemeinsamen Teiler, (Festschrift der Universität Braunschweig, 1897) in Gesammelte mathematische Werke, Band 2.
    Braunschweig: Friedr. Vieweg & Sohn A.-G., 1931.
  • [EG80] P. Erdos and R.L. Graham.
    Old and New Problems and Results in Combinatorial Number Theory.
    Université de Genève, 1980.
    Monographie No. 28 de L'Enseignement Mathématique. MR 82j:10001
  • [Els96] C. Elsholtz.
    The Erdos-Straus conjecture on $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.
    Diploma thesis, Technische Universität Darmstadt, 1996.
  • [Els98] C. Elsholtz.
    Sums of $k$ Unit Fractions.
    PhD thesis, Technische Universität Darmstadt, 1998.
  • [Erd50] P. Erdos.
    Az $\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n} =\frac{a}{b}$ egyenlet egész számú megoldásairól (On a Diophantine equation).
    Mat. Lapok, 1:192-210, 1950. MR 13:208b
  • [Gra64] R.L. Graham.
    On finite sums of unit fractions.
    Proc. London Math. Soc. (3), 14:193-207, 1964. MR 28:3968
  • [Guy94] R.K. Guy.
    Unsolved Problems in Number Theory, second edition.
    Springer-Verlag, 1994. MR 96e:11002
  • [Kar93] A.A. Karatsuba.
    Basic Analytic Number Theory.
    Springer Verlag, 1993. MR 94a:11001
  • [Li81] Delang Li.
    On the Equation $\frac{4}{n}= \frac{1}{x} +\frac{1}{y}+\frac{1}{z}$.
    J. Number Theory, 13:485-494, 1981.
    See also Letter to the editor, J. Number Theory 15:282, 1982. MR 83e:10026; MR 84b:10024
  • [Mar00] G. Martin.
    Denser Egyptian fractions.
    Acta Arith. 95:231-260, 2000.
  • [Mar99] G. Martin.
    Dense Egyptian fractions.
    Trans. Amer. Math. Soc., 351:3641-3657, 1999. MR 99m:11035
  • [Mon78] H.L. Montgomery.
    The analytic principle of the large sieve.
    Bull. Amer. Math. Soc., 84:547-567, 1978. MR 57:5931
  • [Mor69] L.J. Mordell.
    Diophantine Equations, volume 30 of Pure and Applied Mathematics.
    Academic Press, 1969. MR 40:2600
  • [Nak39] M. Nakayama.
    On the Decomposition of a Rational Number into ``Stammbrüche".
    Tôhuku Math. J., 46:1-21, 1939. MR 1:134c
  • [Nar86] W. Narkiewicz.
    Classical Problems in Number Theory, volume 62 of Mathematical Monographs.
    PWN, 1986. MR 90e:11002
  • [San91] J.W. Sander.
    On $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ and Rosser's sieve.
    Acta Arith., 59:183-204, 1991. MR 92j:11031
  • [San94] J.W. Sander.
    On $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ and Iwaniec' Half Dimensional Sieve.
    J. Number Theory, 46:123-136, 1994. MR 95e:11044
  • [San97] J.W. Sander.
    Egyptian Fractions and the Erdos-Straus Conjecture.
    Nieuw Arch. Wisk. (4), 15:43-50, 1997. MR 98d:11039
  • [Scha] A. Schinzel.
    Erdos's work on finite sums of unit fractions.
    To appear in Paul Erdos and his Mathematics, Proceedings of the Erdos conference (Budapest 1999), (Editors: G. Hálasz, L. Lovász, M. Simonovits, V. Sós).
  • [Sch00] A. Schinzel.
    On sums of three unit fractions with polynomial denominators.
    Funct. Approx. Comment. Math. 28:187-194, 2000.
  • [Sch56] A. Schinzel.
    Sur quelques propriétés des nombres $\frac{3}{n}$ et $\frac{4}{n}$, où $n$ est un nombre impair.
    Mathesis, 65:219-222, 1956. MR 18:284a
  • [Sch74] W. Schwarz.
    Einführung in die Siebmethoden der analytischen Zahlentheorie.
    Bibliographisches Institut, Mannheim, 1974. MR 53:13147
  • [She86] Shen Zun.
    On the diophantine equation $\sum_{i=0}^k \frac{1}{x_i} = \frac{a}{n}$.
    Chinese Ann. Math. Ser. B, 7:213-220, 1986. MR 87j:11026
  • [Sie56] W. Sierpinski.
    Sur les décompositions de nombres rationnels en fractions primaires.
    Mathesis, 65:16-32, 1956. MR 17:1185d
  • [Sos05] E. Sós.
    Die diophantische Gleichung $\frac{1}{x}=\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}$.
    Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 36:97-102, 1905.
  • [Sos06] E. Sós.
    Zwei diophantische Gleichungen.
    Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 37:186-190, 1906.
  • [Vau70] R.C. Vaughan.
    On a problem of Erdos, Straus and Schinzel.
    Mathematika, 17:193-198, 1970. MR 44:6600
  • [Vau73] R.C. Vaughan.
    Some Applications of Montgomery's Sieve.
    J. Number Theory, 5:64-79, 1973. MR 49:7222
  • [Vio73] C. Viola.
    On the diophantine equations $\prod_0^k x_i - \sum_0^k x_i=n$ and $\sum_0^k \frac{1}{x_i} = \frac{a}{n}$.
    Acta Arith., 22:339-352, 1973. MR 48:234
  • [Web70] W.A. Webb.
    On $\frac{4}{n}= \frac{1}{x} + \frac{1}{y} +\frac{1}{z}$.
    Proc. Amer. Math. Soc., 25:578-584, 1970. MR 41:1639
  • [Yan82] Xun Qian Yang.
    A note on $\frac{4}{n} = \frac{1}{x} +\frac{1}{y}+ \frac{1}{z}$.
    Proc. Amer. Math. Soc., 85:496-498, 1982. MR 83j:10017

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Additional Information

Christian Elsholtz
Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstrasse 1, D-38678 Clausthal-Zellerfeld, Germany

Received by editor(s): May 23, 2000
Received by editor(s) in revised form: August 28, 2000
Published electronically: April 12, 2001
Additional Notes: The research for this paper was supported by a Ph.D. grant from the German National Merit Foundation
Article copyright: © Copyright 2001 American Mathematical Society

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