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Open problem · Mathematics · since 1948

Erdős–Straus conjecture

Can 4/n be written as 1/x + 1/y + 1/z with positive whole numbers x, y and z for every n ≥ 2?

Summary

Erdős and Straus conjectured in 1948 that every fraction 4/n splits into three fractions with numerator 1. Computer searches have confirmed it for all n up to 10¹⁷, but no general proof is known.

Why it is open

The conjecture appeared in a 1950 paper of Paul Erdős, who attributed it to himself and Ernst G. Straus; when it was first made is not known [5]. A paper by Richard Obláth, submitted in 1948, mentions it as a conjecture of Erdős [1, 5].

Four unit fractions are always enough. The greedy algorithm, which repeatedly takes the largest unit fraction that still fits, writes any fraction m/n between 0 and 1 as a sum of at most m distinct unit fractions; for 4/17 it gives 1/5 + 1/29 + 1/1233 + 1/3039345, although 4/17 = 1/6 + 1/17 + 1/102 [5]. The conjecture says that three always suffice.

A solution for n gives one for every multiple of n, so it is enough to prove the conjecture for primes [1, 5]. And explicit identities settle whole residue classes at once: for example, 4/(8t + 5) = 1/(2(t + 1)) + 1/((t + 1)(8t + 5)) + 1/(2(t + 1)(8t + 5)) for every t [5]. But such formulas cannot cover everything. Andrzej Schinzel proved that no formula of this kind exists for residue classes of quadratic residues (squares modulo the modulus); in particular there is none for all n = 4t + 1 [5, 8]. For an odd prime p, every solution has one of two shapes, with one denominator or two divisible by p, which Christian Elsholtz and Terence Tao call Type I and Type II; neither shape exists for odd perfect squares, although squares have solutions of other shapes [1, 8]. A method that proves such solutions exist for every prime must therefore fail for squares such as p², which, they write, "already rules out many strategies (e.g. a finite set of covering congruence strategies, or the circle method)" [1].

No prize or rule sets the bar: a resolution would be a proof for every prime or a single counterexample. The conjecture is Erdős Problem #242 in Thomas Bloom's database of Erdős problems [9].

What is known

Congruence classes (1948–1971). Obláth noted that the conjecture holds whenever n + 1 is divisible by a prime congruent to 3 modulo 4, which already implies that it holds for almost all n [7, 9]. Bloom and Elsholtz explain the mechanism: if 4acd − 1 = m, every n ≡ −a/c (mod m) has a solution, and the number of residue classes settled in this way modulo 4t − 1 depends essentially on the number of ways to write t as a product of three positive integers, which on average grows like (log t)² [5]. Using such parametric solutions, Louis Mordell proved the conjecture for all n outside six residue classes modulo 840, namely 1, 121, 169, 289, 361 and 529 [6, 9]. Elsholtz and Tao note that every residue class modulo 840 can be solved by polynomial formulas unless it consists of perfect squares, and that classes of squares cannot be [1]. In 1971 D. G. Terzi extended the result to all n outside 198 possible residue classes modulo 120120 [9]. Bloom and Elsholtz give a proof that the conjecture is equivalent to every prime lying in at least one of two explicit families of residue classes, noting that similar statements appear in earlier work [5].

Density and counting (1970–2020). R. C. Vaughan combined the soluble classes with the large sieve and proved that at most N exp(−c(log N)^(2/3)) numbers up to N can be exceptions, for some constant c > 0 [1, 5]. Writing in 2025, Carl Pomerance and Andreas Weingartner note that the count of possible exceptions "has been strongly improved, though not recently", citing Vaughan's 1970 bound [7]. Writing f(n) for the number of solutions, Elsholtz and Tao proved that the sum of f(p) over primes p up to N lies between constant multiples of N log² N and N log² N log log N, so that a typical prime has only O(log³ p log log p) solutions [1]. They also proved that f(p) ≤ p^(3/5 + o(1)) for every prime p [1, 9]. Elsholtz and Stefan Planitzer extended a bound of this size to every n for a fixed numerator m [3], and proved that f(n) ≥ (log n)^(log 6 + o(1)) for almost all n [5].

Computation (1999–2025). Allan Swett checked the conjecture up to 10¹⁴ in 1999 with a sieve built on a single modular equation [2]. In 2014 Serge Salez proved that a set of seven modular equations, three of them new, is "complete", and checked all n up to 10¹⁷ with an optimised sieve in about 16 hours of computing [1, 2]. In August 2025 Spiridon Mihnea and Bogdan C. Dumitru extended Salez's method with one more modular filter and reported checking all primes up to 10¹⁸ in about two weeks [6]. Their four-page paper is a preprint, not peer reviewed; erdosproblems.com cites it for the 10¹⁸ bound [6, 9].

Geometry (2020). Martin Bright and Daniel Loughran recast the conjecture as a question about integer points on an algebraic surface, an affine cubic of the kind called log K3 [4]. They proved that for every n ≥ 2 there is no Brauer–Manin obstruction to natural-number solutions; such an obstruction, when present, rules solutions out [4]. The same method gives new conditions that every natural-number solution must satisfy and recovers Elsholtz and Tao's result on odd squares [4].

Recent work (2025–2026). Schinzel conjectured that for every numerator m ≥ 4, m/n is a sum of three unit fractions for all sufficiently large n [5]. In a preprint (version 2, January 2026), Pomerance and Weingartner proved that if such a threshold exists it must be at least exp(m^(1/3 + o(1))), and that for every m ≥ 6.52 × 10⁹ some prime p between m² and 2m² is an exception; their computations support the second statement for m ≥ 20 [7]. In April 2026 Marc Chamberland showed in Integers that, for a prime p, 4/p has a Type II solution (exactly two denominators divisible by p) if and only if p = qr − 4s₁s₂ for positive integers q, r, s₁, s₂ with q ≡ 3 (mod 4) and s₁, s₂ dividing (q + 1)/4; he conjectures that all primes can be written in this form [8].

Main approaches

Covering congruences and computation. Find enough identities that every prime falls into at least one solved residue class, an aim Bloom and Elsholtz show is equivalent to the conjecture [5]. Strongest evidence: Mordell's and Terzi's classes [9], and the modular sieves that verified the conjecture up to 10¹⁷, and in a preprint 10¹⁸ [2, 6]. Main difficulty: no polynomial formula solves a class of quadratic residues [5, 8], and Elsholtz and Tao count a finite set of covering congruences among the strategies that the odd-square obstruction rules out [1].

Sieves and probabilistic heuristics. Show that exceptions are rare, then aim to exclude them. Strongest evidence: Vaughan's bound [1, 5], and Bloom and Elsholtz's heuristic that so many soluble classes leave at most finitely many exceptions, which with the computations "strongly suggests the conjecture holds for all n" [5]. Main difficulty: Elsholtz and Tao stress that such a model "is only a heuristic and does not constitute a rigorous argument" [1], and the bound on exceptions that Pomerance and Weingartner cite dates from 1970 [7].

Counting solutions. Bound the number of solutions f(p) from below for every prime. Strongest evidence: averages over primes and lower bounds for almost all n [1, 3, 5]. Main difficulty: a typical prime has only about log³ p solutions, up to a factor log log p, and Elsholtz and Tao suggest that this may be why methods such as the circle method, which usually need polynomially many solutions, appear insufficient [1].

Arithmetic geometry. Treat the equation as a problem of integer points on a surface and use obstructions from algebraic geometry [4]. Strongest evidence: the absence of a Brauer–Manin obstruction for every n ≥ 2 [4]. Main difficulty: the authors present the result as showing that this obstruction cannot disprove the conjecture, and they also prove that on these surfaces not every failure of strong approximation, a related local-to-global property, is explained by the Brauer–Manin obstruction [4].

Open sub-questions

  1. The six classes modulo 840. Which further identities, beyond Terzi's classes modulo 120120, most reduce the primes in the classes 1, 121, 169, 289, 361 and 529 modulo 840 that a verification must search directly [2, 6, 9]?
  2. Vaughan's bound. Can the bound N exp(−c(log N)^(2/3)) on the number of exceptions up to N, from 1970, be improved [7]?
  3. Type II for every prime. Can every prime p be written as qr − 4s₁s₂ with q ≡ 3 (mod 4) and s₁, s₂ dividing (q + 1)/4, as Chamberland conjectures [8]?
  4. Fewest solutions. For the primes in the six hard classes up to 3.5 × 10⁷, whose solutions Mihnea and Dumitru counted, does the minimum of f(p) grow like the log³ p intensity of Elsholtz and Tao's heuristic model [1, 6]?
  5. Schinzel's threshold. Pomerance and Weingartner's heuristic predicts no exceptional primes above exp(m^(1/2 + ε)) for large m; can any finite threshold be proved for some numerator m ≥ 5 [7]?

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Key references

  1. Elsholtz, C. & Tao, T. (2013). Counting the number of solutions to the Erdős–Straus equation on unit fractions. Journal of the Australian Mathematical Society 94(1), 50–105. 10.1017/S1446788712000468 · Checked 15 Oct 2026
  2. Salez, S. E. (2014). The Erdős–Straus conjecture: new modular equations and checking up to N = 10¹⁷. Preprint, arXiv:1406.6307 (version 1, 24 June 2014), not peer reviewed. Source · Checked 15 Oct 2026
  3. Elsholtz, C. & Planitzer, S. (2020). The number of solutions of the Erdős–Straus equation and sums of k unit fractions. Proceedings of the Royal Society of Edinburgh Section A: Mathematics 150(3), 1401–1427 (published online 2019). 10.1017/prm.2018.137 · Checked 15 Oct 2026
  4. Bright, M. & Loughran, D. (2020). Brauer–Manin obstruction for Erdős–Straus surfaces. Bulletin of the London Mathematical Society 52(4), 746–761. 10.1112/blms.12374 · Checked 15 Oct 2026
  5. Bloom, T. F. & Elsholtz, C. (2022). Egyptian fractions. Nieuw Archief voor Wiskunde (5) 23(4), 237–245. Source · Checked 15 Oct 2026
  6. Mihnea, S. & Dumitru, B. C. (2025). Further verification and empirical evidence for the Erdős–Straus conjecture. Preprint, arXiv:2509.00128 (version 1, 29 August 2025), not peer reviewed. Source · Checked 15 Oct 2026
  7. Pomerance, C. & Weingartner, A. (2025). Exceptions to the Erdős–Straus–Schinzel conjecture. Preprint, arXiv:2511.16817 (version 2, 15 January 2026), not peer reviewed. Source · Checked 15 Oct 2026
  8. Chamberland, M. (2026). The Erdős–Straus conjecture and the structure of primes. Integers 26, #A42 (published 3 April 2026). 10.5281/zenodo.19403738 · Checked 15 Oct 2026
  9. Bloom, T. F. Erdős Problem #242. erdosproblems.com, last edited 7 May 2026, accessed 2026-10-15. Source · Checked 15 Oct 2026

Page reviewed 15 Oct 2026

Source

Open, as listed by Wikipedia · Unsolved problems in mathematics. Checked 08 Oct 2026.

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