A note on the Erdős–Straus conjecture

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• 作者：S. Subburam ; Alain Togbé
• 关键词：Egyptian fractions ; Diophantine equation ; Number of solutions
• 刊名：Periodica Mathematica Hungarica
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：72
• 期：1
• 页码：43-49
• 全文大小：382 KB
• 参考文献：1.T. Browning, C. Elsholtz, The number of representations of rationals as a sum of unit fractions. Ill. J. Math. 55(2), 685–696 (2011)
  2.Y.-G. Chen, C. Elsholtz, L.-L. Jiang, Egyptian fractions with restrictions. Acta Arith. 154, 109–123 (2012)MathSciNet CrossRef MATH
  3.C. Elsholtz, Sums of \(k\) unit fractions. Trans. Am. Math. Soc. 353, 3209–3227 (2001)MathSciNet CrossRef MATH
  4.C. Elsholtz, T. Tao, Counting the number of solutions to the Erdős–Straus equation on unit fractions. J. Aust. Math. Soc. 94, 50–105 (2013)
  5.P. Erdős, Az \(1/x_1 + 1/x_2 + \cdots + 1/x_n = a/b\) egyenlet egész számú megoldásairól. Mat. Lapok 1, 192–210 (1950)MathSciNet
  6.R. Guy, Unsolved Problems in Number Theory, 2nd edn. (Springer-Verlag, New York, 1994), pp. 158–166CrossRef
  7.C. Jia, A Note on Terence Tao’s Paper “On the Number of Solutions to \(4/p = 1/n_1 + 1/n_2 + 1/n_3\) ”, preprint
  8.C. Jia, The estimate for mean values on prime numbers relative to \(4/p = 1/n_1 + 1/n_2 + 1/n_3\) . Sci. China Math. 55(3), 465–474 (2012)MathSciNet CrossRef
  9.D. Li, On the Equation \(4/n = 1/x + 1/y + 1/z\) . J. Number Theory 13, 485–494 (1981)MathSciNet CrossRef MATH
  10.M. Nakayama, On the Decomposition of a rational number into “Stammbrüche”. Tohoku Math. J. 46, 1–21 (1939)MathSciNet MATH
  11.M.R. Obláth, Sur l’ équation diophantienne \(4/n = 1/x_1 + 1/x_2 + 1/x_3\) . Mathesis 59, 308–316 (1950)MathSciNet MATH
  12.J.W. Sander, On \(4/n = 1/x + 1/y + 1/z\) and Rosser’s sieve. Acta Arith. 59, 183–204 (1991)MathSciNet MATH
  13.J.W. Sander, On \(4/n = 1/x+1/y+1/z\) and Iwaniec’ Half Dimensional Sieve. J. Number Theory 46, 123–136 (1994)MathSciNet CrossRef MATH
  14.A. Schinzel, Sur quelques propriétés des nombres \(3\) et \(4\) , où \(n\) est un nombre impair. Mathesis 65, 219–222 (1956)MathSciNet MATH
  15.W. Sierpiński, Sur les décompositions de nombres rationelles en fractions primaires. Mathesis 65, 16–32 (1956)MathSciNet
  16.S. Subburam, R. Thangadurai, On the Diophantine equation \(x^{3} + by + 1 -xyz = 0\) . C.R. Math. Rep. Acad. Sci. Canada 36(1), 15–19 (2014)MathSciNet MATH
  17.R. Vaughan, On a problem of Erdős, Straus and Schinzel. Mathematika 17, 193–198 (1970)MathSciNet CrossRef MATH
  18.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)MathSciNet
  19.W. Webb, On \(4/n = 1/x + 1/y + 1/z\) . Proc. Am. Math. Soc. 25, 578–584 (1970)MATH
  20.W. Webb, On a theorem of Rav concerning Egyptian fractions. Can. Math. Bull. 18(1), 155–156 (1975)MathSciNet CrossRef MATH
  21.W. Webb, On the Diophantine equation \(k/n = a_1/x_1 +a_2/x_2 +a_3/x_3\) . Časopis pro pěstováni matematiy, roč 101, 360–365 (1976)
  22.K. Yamamoto, On the Diophantine Equation \(4/n = 1/x + 1/y + 1/z\) . Mem. Fac. Sci. Kyushu Univ. Ser. A 19, 37–47 (1965)MathSciNet MATH
  23.X.Q. Yang, A note on \(4/n = 1/x + 1/y + 1/z\) . Proc. Am. Math. Soc. 85, 496–498 (1982)MATH
  
• 作者单位：S. Subburam (1)
  Alain Togbé (2)
  
  1. Department of Mathematics, Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai, 600 113, India
  2. Department of Mathematics, Purdue University North Central, 1401 S, U.S. 421, Westville, IN, 46391, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Sciences
  Mathematics
  
• 出版者：Akad茅miai Kiad贸, co-published with Springer Science+Business Media B.V., Formerly Kluwer Academic
• ISSN：1588-2829

文摘

In this note, we consider the Erdős–Straus Diophantine equation $$\begin{aligned} \frac{c}{n}=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}, \end{aligned}$$where n and c are positive integers with \(\gcd (n, c) = 1\). We provide a formula for the number f(n, c) of all positive integral solutions (x, y, z) of the equation. In 1948, Erdős and Straus conjectured that \(f(n,4) \ge 1,\) for all integers \(n \ge 2\). Here, we solve the conjecture for a special case of n. Keywords Egyptian fractions Diophantine equation Number of solutions Mathematics Subject Classification 11D68 11D72 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (23) References1.T. Browning, C. Elsholtz, The number of representations of rationals as a sum of unit fractions. Ill. J. Math. 55(2), 685–696 (2011)2.Y.-G. Chen, C. Elsholtz, L.-L. Jiang, Egyptian fractions with restrictions. Acta Arith. 154, 109–123 (2012)MathSciNetCrossRefMATH3.C. Elsholtz, Sums of \(k\) unit fractions. Trans. Am. Math. Soc. 353, 3209–3227 (2001)MathSciNetCrossRefMATH4.C. Elsholtz, T. Tao, Counting the number of solutions to the Erdős–Straus equation on unit fractions. J. Aust. Math. Soc. 94, 50–105 (2013)5.P. Erdős, Az \(1/x_1 + 1/x_2 + \cdots + 1/x_n = a/b\) egyenlet egész számú megoldásairól. Mat. Lapok 1, 192–210 (1950)MathSciNet6.R. Guy, Unsolved Problems in Number Theory, 2nd edn. (Springer-Verlag, New York, 1994), pp. 158–166CrossRef7.C. Jia, A Note on Terence Tao’s Paper “On the Number of Solutions to \(4/p = 1/n_1 + 1/n_2 + 1/n_3\)”, preprint8.C. Jia, The estimate for mean values on prime numbers relative to \(4/p = 1/n_1 + 1/n_2 + 1/n_3\). Sci. China Math. 55(3), 465–474 (2012)MathSciNetCrossRef9.D. Li, On the Equation \(4/n = 1/x + 1/y + 1/z\). J. Number Theory 13, 485–494 (1981)MathSciNetCrossRefMATH10.M. Nakayama, On the Decomposition of a rational number into “Stammbrüche”. Tohoku Math. J. 46, 1–21 (1939)MathSciNetMATH11.M.R. Obláth, Sur l’ équation diophantienne \(4/n = 1/x_1 + 1/x_2 + 1/x_3\). Mathesis 59, 308–316 (1950)MathSciNetMATH12.J.W. Sander, On \(4/n = 1/x + 1/y + 1/z\) and Rosser’s sieve. Acta Arith. 59, 183–204 (1991)MathSciNetMATH13.J.W. Sander, On \(4/n = 1/x+1/y+1/z\) and Iwaniec’ Half Dimensional Sieve. J. Number Theory 46, 123–136 (1994)MathSciNetCrossRefMATH14.A. Schinzel, Sur quelques propriétés des nombres \(3\) et \(4\), où \(n\) est un nombre impair. Mathesis 65, 219–222 (1956)MathSciNetMATH15.W. Sierpiński, Sur les décompositions de nombres rationelles en fractions primaires. Mathesis 65, 16–32 (1956)MathSciNet16.S. Subburam, R. Thangadurai, On the Diophantine equation \(x^{3} + by + 1 -xyz = 0\). C.R. Math. Rep. Acad. Sci. Canada 36(1), 15–19 (2014)MathSciNetMATH17.R. Vaughan, On a problem of Erdős, Straus and Schinzel. Mathematika 17, 193–198 (1970)MathSciNetCrossRefMATH18.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)MathSciNet19.W. Webb, On \(4/n = 1/x + 1/y + 1/z\). Proc. Am. Math. Soc. 25, 578–584 (1970)MATH20.W. Webb, On a theorem of Rav concerning Egyptian fractions. Can. Math. Bull. 18(1), 155–156 (1975)MathSciNetCrossRefMATH21.W. Webb, On the Diophantine equation \(k/n = a_1/x_1 +a_2/x_2 +a_3/x_3\). Časopis pro pěstováni matematiy, roč 101, 360–365 (1976)22.K. Yamamoto, On the Diophantine Equation \(4/n = 1/x + 1/y + 1/z\). Mem. Fac. Sci. Kyushu Univ. Ser. A 19, 37–47 (1965)MathSciNetMATH23.X.Q. Yang, A note on \(4/n = 1/x + 1/y + 1/z\). Proc. Am. Math. Soc. 85, 496–498 (1982)MATH About this Article Title A note on the Erdős–Straus conjecture Journal Periodica Mathematica Hungarica Volume 72, Issue 1 , pp 43-49 Cover Date2016-03 DOI 10.1007/s10998-015-0109-9 Print ISSN 0031-5303 Online ISSN 1588-2829 Publisher Springer Netherlands Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords Egyptian fractions Diophantine equation Number of solutions 11D68 11D72 Authors S. Subburam (1) Alain Togbé (2) Author Affiliations 1. Department of Mathematics, Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai, 600 113, India 2. Department of Mathematics, Purdue University North Central, 1401 S, U.S. 421, Westville, IN, 46391, USA Continue reading... To view the rest of this content please follow the download PDF link above.