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ON THE DIOPHANTINE EQUATION $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$
Keywords:
Egyptian fraction, Erdős-Strauss conjecture, Fibonacci-Sylvester algorithm, integral solutionDOI:
https://doi.org/10.17654/0972555524028Abstract
Based on the Fibonacci-Sylvester algorithm, we introduce a new elementary method in terms of the Fibonacci-Sylvester algorithm for studying the Erdos-Straus conjecture (ESC), that is, the case of all positive integer solutions of the Diophantine equation $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$. Here, only elementary methods are used to provide the general solution expressions for its all positive integer solutions. Using this new method, we provide a new proof of the Mordell theorem, which states that $\frac{4}{n}$ has a expression as the sum of three unit fractions for every natural number $n$, except possibly for the numbers of the form $n \equiv u(\bmod 840)$ with $u=1,11^2, 13^2, 17^2, 19^2, 23^2$. In addition, we also explore the situation of the Erdos-Straus conjecture when $n(\bmod 1320)$ and $n(\bmod 9240)$, and give the corresponding general solutions for all positive integer solutions of the equation, which is helpful for solving the problem of ESC and promoting the related research.
Received: April 20, 2024
Revised: July 15, 2024
Accepted: August 10, 2024
References
os, Az egyenlet egesz szamu megoldasairol (on a Diophantine equation), Mat. Lapok 1 (1950), 192-210.
Z. Ke and Q. Sun, Some conjecture and problem in number theory, Chinese Journal of Nature (in Chinese), 7 (1979), 411-413.
Yamamoto, On the Diophantine equations K. Men. Fac. Sci. Kyushu University, Ser. A., 19 (1965), 37-47.
N. Franceschine, Egyptian Fractions, Sonoma State Coll., 1978.
A. Swett, The Erdos-Straus conjecture, Rev. 10 (1999), 28-99.
S. E. Salez, The Erdos-Straus conjecture new modular equations and checking up to arxiv preprint arxiv:1406.6307 (2014).
L. J. Mordell, Diophantine Equations, Academic Press, London/New York, 1969.
L. A. Rosati, On the Diophantine equation (English), Sull’equazione diofantea (original in Italian), Bollettino Della Unione Matematica Italiana, 3 (1954), 59-63.
R. C. Vaughan, On a problem of Erdos-Straus and Scinzel, Mathematica 17 (1970), 193-198.
R. K. Guy, Egyptian Fractions, §D11 in Unsolved Problems in Number Theory, 2nd ed., New York: Springer-Verlag, 1994, pp. 158-166.
M. H. Ahmadi and M. N. Bleicher, On the conjectures of Erdos and Straus, and Sipinski on Egyptian fractions, International Journal of Mathematical and Statistical Sciences 7(2) (1998), 169-185.
L. Bernstein, To the solution of the Diophantine equation particularly in the case of the (English), Zur Losung der diophantischen Gieichung insbesondere im Fall Journal fur die Reine und Angewandte Mathematik 211 (1962), 1-10.
D. G. Terzi, On a conjecture by Erdos-Straus, Nordisk Tidskr, Informations- Behandling (BIT), 11 (1971), 212-216.
M. B. Crawford, On the Number of Representations of One as the Sum of Unit Fractions, Masters of Science in Mathematics, Virginia Polytechnic Institute and State University, 2019.
M. D. Giovanni, S. Gallipoli and M. Gionfriddo, Historical origin and scientific development of graphs, hypergraphs and design theory, Bulletin of Mathematics and Statistics Research 7 (2019), 19-23.
C. Elsholtz and T. Tao, Counting the number of solutions to Erdos-Straus equations on unit fractions, J. of Australasian Math. Society 94(1) (2013), 50-105.
J. Ghanouchi, An analytic approach of some conjectures related to Diophantine equations, Bulletin of Mathematical Sciences and Applications 1 (2012), 29-40.
J. Ghanouchi, About the Erdos conjecture, International Journal of Science and Research 4(2) (2015), 341-341.
D. J. Negash, Solutions to Diophantine equation of Erdos-Straus conjecture, arxiv preprint arxiv.1812.05684 (2018).
J. W. Sander, and Iwaniec’ half dimensional sieve, J. of Number Theory 46 (1994), 123-136.
M. Dunton and R. E. Grimm, Fibonacci on Egyptian Fractions, Fibonacci Quarterly 4 (1966), 339-354. Translation of Fibonacci’s original work, Liber Abacci (1202).
J. J. Sylvester, On a point in the theory of vulgar fractions, Amer. J. Math. 3 (1880), 332-335.
W. A. Webb, On the Diophantine equation Casopis pro pestovanl matematiky 101(4) (1967), 360-365.
E. S. Croot III, Egyptian Fractions, Ph. D. Thesis, 1994.
G. G. Martin, The distribution of prime primitive roots and dense Egyptian fractions, Ph. D. Thesis, 1997.
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