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Erdős-Straus Conjecture

The Erdős-Straus conjecture, due to Paul Erdős and E. G. Straus, states that the Diophantine equation

4/n=1/a+1/b+1/c

involving Egyptian fractions always can be solved (Obláth 1950, Rosati 1954, Bernstein 1962, Yamamoto 1965, Vaughan 1970, Guy 1994). Swett has established validity of the conjecture for all n<=10^(14). The conjecture is problem 242 in the Erdős problems collection (Bloom 2026).

See also

Egyptian Fraction, Erdős Problems

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References

Bernstein, L "Zur Lösung der diophantischen Gleichung m/n=1/x+1/y+1/z insbesondere im Falle m=4." J. reine angew. Math. 211, 1-10, 1962.Bloom, T. F. "Erdős Problem 242." Erdős Problems. Oct. 1, 2026. https://www.erdosproblems.com/242.Guy, R. K. "Egyptian Fractions." §D11 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 158-166, 1994.Obláth, R. "Sur l'equation diophantienne 4/n=1/x_1+1/x_2+1/x_3." Mathesis 59, 308-316, 1950.Rosati, L. A. "Sull'equazione diofantea 4/n=1/x_1+1/x_2+1/x_3." Boll. Un. Mat. Ital. 9, 59-63, 1954.Swett, A. "The Erdos-Strauss Conjecture." Rev. 10/28/99. http://math.uindy.edu/swett/esc.htm.Vaughan, R. C. "On a Problem of Erdős, Straus and Schinzel." Mathematika 17, 193-198, 1970.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 29, 1986.Yamamoto, K. "On the Diophantine Equation 4/n=1/x+1/y+1/z." Mem. Fac. Sci. Kyushu U. Ser. A 19, 37-47, 1965.

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Erdős-Straus Conjecture

Cite this as:

Weisstein, Eric W. "Erdős-Straus Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Erdos-StrausConjecture.html

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