The Erdős–Straus conjecture is an unsolved problem in elementary number theory concerning the representation of fractions of the form $ \frac{4}{n} $ as a sum of three unit fractions, also known as Egyptian fractions. Formally, the conjecture states that for every integer $ n \ge 2 $ there exist positive integers $ x, y, z $ such that
$$ \frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}. $$
Historical background
The conjecture was first posed by Paul Erdős and Ernst G. Straus in 1948. It arose from their interest in Diophantine equations and the study of Egyptian fraction representations, a topic with a long history dating back to ancient mathematics.
Known results
- The conjecture has been verified computationally for all integers $ n $ up to very large bounds; as of the latest published computational checks, it holds for all $ n \le 10^{17} $ (the exact bound may be updated in newer literature).
- Certain infinite families of integers satisfy the conjecture by explicit constructions. For example, if $ n $ is a multiple of 4, a simple solution is obtained by setting $ x = n, y = n/2, z = n/4 $.
- Various partial results have been obtained using analytic and combinatorial methods. Notably, Vaughan (1970) showed that the conjecture holds for all sufficiently large $ n $ outside a set of density zero, and subsequent refinements have reduced the exceptional set further.
- No counterexample has been found, and the conjecture remains open for the general case.
Mathematical significance
The Erdős–Straus conjecture is an example of a problem that is easy to state but has resisted proof for more than seven decades. It connects to broader themes in additive number theory, such as the representation of rational numbers as sums of unit fractions, and it motivates research into the distribution of solutions to Diophantine equations.
Current status
The conjecture remains unproved. Research continues both in extending computational verification to larger ranges and in developing theoretical approaches that might resolve the conjecture for all integers $ n \ge 2 $. No consensus exists on a likely pathway to a full proof or disproof.