Erdos-Straus Conjecture

Paul Erdős

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Born: 26 March 1913, Budapest, Austria-Hungary.

Died:  20 September 1996 (aged 83), Warsaw, Poland.

Ernst G. Straus

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Born: 25 February 1922 , Munich, Germany.

Died:  12 July 1983 (aged 61), Los Angeles, California, U.S.

Introduction: Unsolved problem in mathematics

The Erdös-Straus conjecture states that the equation 4n=1x+1y+1z has positive integer solutions x,y,z for every postive integers n that is 2 or more.n≥

One important topic in number theory is the study of Diophantine equations, equations in which only integer solutions are permitted.

Example.

For n=2, \begin{equation} \frac{4}{2}=\frac{1}{2}+\frac{1}{2}+\frac{1}{1} \end{equation}

For n=3, \begin{equation} \frac{4}{3}=\frac{1}{1}+\frac{1}{4}+\frac{1}{12} \end{equation}

For n=4, \begin{equation} \frac{4}{4}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \end{equation}

For n=5, \begin{equation} \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20} \end{equation}

References

1  Bernstein, Leon (1962), "Zur Lösung der diophantischen Gleichung ${\displaystyle �/�=1/�+1/�+1/�}$, insbesondere im Fall ${\displaystyle �=4}$", Journal für die Reine und Angewandte Mathematik (in German), 211: 1–10, doi:10.1515/crll.1962.211.1, MR 0142508, S2CID 118098315

2  Kotsireas, Ilias (1999), "The Erdős-Straus conjecture on Egyptian fractions", Paul Erdős and his mathematics (Budapest, 1999), Budapest: János Bolyai Math. Soc., pp. 140–144, MR 1901903