Erdős–Straus conjecture

Erdős–Straus conjecture
Name	Erdős–Straus conjecture
Field	Number theory
Conjectured by	Paul Erdős; Ernst G. Straus
Year	1948
Status	Open (as of 2026)

Erdős–Straus conjecture The Erdős–Straus conjecture posits that for every integer n ≥ 2 the rational number 4/n can be expressed as a sum of three positive unit fractions. This conjecture is a problem in analytic and additive Number theory closely connected to classical investigations by Diophantus of Alexandria and modern work by Paul Erdős and Ernst G. Straus. The question sits at the intersection of research pursued at institutions such as Princeton University, University of Cambridge, and ETH Zurich and draws methods from research traditions exemplified by Carl Friedrich Gauss, Leonhard Euler, and Srinivasa Ramanujan.

Statement

The conjecture asserts: for every integer n ≥ 2 there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. This formulation connects to classical Egyptian fraction representations studied by Diophantus of Alexandria and later by Fibonacci and Johann Faulhaber, and it invites techniques used by researchers at Institute for Advanced Study and Max Planck Institute for Mathematics influenced by work of Paul Erdős and Alfréd Rényi.

History and progress

The conjecture was proposed in 1948 by Paul Erdős and Ernst G. Straus and has since motivated contributions from mathematicians affiliated with University of California, Berkeley, Harvard University, University of Oxford, and École Polytechnique. Early partial results echo work by Leonhard Euler on Egyptian fractions and by Dirichlet on arithmetic progressions, while analytic techniques later employed reflect traditions from G. H. Hardy and John Littlewood. Subsequent developments involved collaborations and independent advances by researchers connected to Bell Labs, Los Alamos National Laboratory, and research groups around Terence Tao, Ben Green, and Henryk Iwaniec.

Known results and partial proofs

Partial results include reduction to special residue classes studied using methods inspired by Dirichlet's theorem on arithmetic progressions and by techniques from sieve theory associated with Atle Selberg and Heath-Brown. Results prove the conjecture for all n in many congruence classes, following strategies related to work by Deshouillers, Elsholtz, Tao, and Schinzel. Researchers at University of Toronto, Université Paris-Sud, and ETH Zurich have shown validity for n up to large bounds contingent on properties reminiscent of results by Vinogradov and Bombieri. Connections have been drawn to classical theorems of Ramanujan and to combinatorial methods developed by Erdős and S\'arközy.

Computational verification

Extensive computation has verified the conjecture for all n up to very large thresholds using techniques employed by teams at NASA, Google Research, Microsoft Research, and university groups at University of Cambridge and University of Oxford. Implementations often use algorithms inspired by work at MIT and Caltech and leverage computational number theory libraries patterned after tools from SageMath and projects associated with Andrew Wiles-era modular methods. Distributed computation projects involving volunteers from SETI-style networks and university clusters have extended verification ranges, reflecting computational paradigms used in the proof of the Four Color Theorem and in computations for the Riemann hypothesis verifications.

Related conjectures and generalizations

The conjecture relates to broader Egyptian fraction problems such as the Sylvester sequence investigations, the Egyptian fraction conjecture variants considered by Erdős and Graham, and generalizations replacing 4 by other integers leading to problems akin to work by Paul Erdős with R. R. Hall. It connects to the study of unit fraction decompositions in contexts explored by Srinivasa Ramanujan and to additive problems reminiscent of the Goldbach conjecture and research by Vinogradov and Goldston.

Methods and approaches

Approaches combine elementary manipulations with algebraic identities used since Diophantus of Alexandria and analytic techniques from the traditions of Hardy and Littlewood. Modern work applies combinatorial number theory from schools led by Paul Erdős and sieve methods influenced by Brun and Selberg, together with computational heuristics developed by researchers at Princeton University and Institute for Advanced Study. Algebraic number theory tools reflecting insights from Emmy Noether and Richard Dedekind sometimes inform structural reductions to special congruence classes.

Open problems and current research directions

Open questions include proving the conjecture in full, finding density results for solutions (drawing on methods by Erdős and Turán), understanding minimal parameterizations related to work by Diophantus of Alexandria and Fibonacci, and extending verification techniques using distributed computing models akin to projects at CERN and Berkeley Lab. Current research by groups at Princeton University, University of Cambridge, ETH Zurich, and University of Chicago explores new sieve-like reductions, algebraic parametrizations, and computational heuristics inspired by progress on problems such as the Twin Prime Conjecture and the ABC conjecture.

Category:Number theory conjectures