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 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Sums of $k$ unit fractions

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

Abstract:

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:

[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
Email: elsholtz@math.tu-clausthal.de

DOI: 10.1090/S0002-9947-01-02782-9
PII: S 0002-9947(01)02782-9
Received by editor(s): May 23, 2000
Received by editor(s) in revised form: August 28, 2000
Posted: April 12, 2001
Additional Notes: The research for this paper was supported by a Ph.D. grant from the German National Merit Foundation
Copyright of article: Copyright 2001, American Mathematical Society




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