For all positive integers \(n \geq 2\), the rational number \(\frac{4}{n}\) can be expressed as the sum of three positive unit fractions.
~Equivalently~
For every integer \(n\geq 2\) there exist positive integers \(x,y,z\) such that
$$\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
| Author | Date Published | |
| Paul Erdős and Ernst G. Straus | 1948 |
A counterexample would have to be a prime number restricted to one of six infinite arithmetic progressions modulo \(840\) [1].
Computer searches have not obtained a counterexample for \(n \leq 10^{17}\) [3]
The number of distinct solutions for \(n\) is given by the sequence OEIS: A073101 and grows irregularly. Elsholtz and Tao [4] showed in 2013 that the average number of solutions for \(n\) is upper bounded polylogarithmically in \(n\).
Related Problems:
[1] Mordell, Louis J. (1967), Diophantine Equations, Academic Press, pp. 287–290
[2] Erdős, Paul (1950), "Az \(1/x1 + 1/x2 + ... + 1/xn = a/b\) egyenlet egész számú megoldásairól (On a Diophantine Equation)" (PDF), Mat. Lapok. (in Hungarian), 1: 192–210, MR 0043117 .
[3] Salez, Serge E. (2014), The Erdős-Straus conjecture New modular equations and checking up to N = 1017, arXiv:1406.6307, Bibcode:2014arXiv1406.6307S
[4] Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős-Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397
Erdős–Straus Conjecture; Egyptian Fractions;