An improved exceptional-set bound for the Erdős

Abstract

Let \(E(N)\) be the number of integers \(2\le n\le N\) for which \(4/n=1/x+1/y+1/z\) has no solution in positive integers. Vaughan proved \(E(N)\ll N\exp\{-c(\log N)^{2/3}\}\). We prove

\[E(N)\ll N\exp\left\{-c(\log N)^{3/4}\right\}.\]

The improvement comes from the Type II divisor fan: for each \(a\ge1\), every divisor of \(a^2\) supplies a congruence class modulo \(4a-1\) of representable \(n\), and the classical representing congruences are the boundary cases of this family. A conductor-separated subfan keeps a mass of order \((\log X)^3\), and a finite-interval sieve turns this mass into the stated saving at the scale \(\log X\asymp(\log N)^{1/4}\).

We also record consequences for prime denominators, for propagation along large prime factors, and for the Erdős–Straus–Schinzel problem with fixed numerator \(m\ge2\), together with a geometric interpretation on the associated log K3 surfaces. Variants with growing numerator or at Elliott–Halberstam strength are stated under explicit additional hypotheses.

2020 Mathematics Subject Classification. 11D68, 11N36, 11N37.

Keywords. Erdős–Straus conjecture, Egyptian fractions, divisor fans, exceptional sets, sieve methods, Bombieri–Vinogradov theorem.

Section 1

Introduction and main theorem

The Erdős–Straus conjecture asserts that for every integer \(n\ge2\) the equation

\[\frac4n=\frac1x+\frac1y+\frac1z\]

has a solution in positive integers. Let \(E(N)\) count the exceptions \(2\le n\le N\). Vaughan proved the classical estimate \(E(N)\ll N\exp\{-c(\log N)^{2/3}\}\) for some absolute \(c>0\). Our main result improves the exponent. The same construction also recovers a logarithmic factor that a one-rough-coordinate sieve loses, through the small-prime saturation device described in Remark 12.1.

Theorem 1.1 (main theorem)

There is an absolute constant \(c>0\) such that, for all \(N\ge3\),

\[E(N)\ll N\exp\left\{-c(\log N)^{3/4}\right\}.\]

The proof covers residue classes by congruence certificates. Each selected certificate gives a residue class modulo some \(q\), and every integer in that class has an explicit representation. The difficulty is to keep cubic total weight while making the residual intersections sparse enough for Brun–Suen: small-prime factors are absorbed into the base class, while conductor separation keeps a cubic mass of residual events.

We estimate the number of integers missed by all selected classes.

Read the introduction and theorem in the paper

Section 2

The Type II divisor fan

The structural starting point is the complete Type II divisor fan

\[4a-1\mid n+4h,\qquad h\mid a^2.\]

The three traditional boundary rays \(h=1,a,a^2\) recover the classical shifts appearing in the work of Mordell, Yamamoto, and Webb, but the interior divisors carry cubic logarithmic mass.

We also prove the fixed-modulus no-obstruction statement for this full cubic fan: for every fixed odd modulus \(M\) and every reduced class \(c\pmod M\), the interior fan supplies cubic mass in the class compatible with \(c\). Thus the local congruences singled out by the boundary tests are not local obstructions to the full fan.

Fixed slices are already classical in the sense of Elsholtz–Tao. Here many interior divisors are used simultaneously, with conductor separation controlling the dependency graph.

Read Section 2 and the certificate proofs

\(a\)12

\(4a-1\): 47
\(h\mid a^2\): 15 divisors
\(h\): 1

n ≡ 43 (mod 47)

Lemma 2.3

Divisor-fan certificate

Let \(n\ge1\), let \(a\ge1\), let \(q=4a-1\), and let \(e\) be a positive divisor of \(a^2\). If

\[n\equiv-4e\pmod q,\]

then \(4/n\) is a sum of three positive unit fractions.

n a e

Sections 4–12

Proof structure

The proof here keeps Vaughan's exceptional-set-covering viewpoint, but changes the certificate family. The dense divisor fan is thinned to a conductor-separated subfan, while the small-prime part of the medium coordinate is saturated. This keeps the logarithmic mass that a one-rough-coordinate sieve would lose.

Section 4

Cubic fan mass and the pairwise-coprime barrier

The full interior fan has cubic logarithmic mass.

\[\sum_{a\le L}\frac{|\Lambda_{\rm int}(a)|}{4a-1}\sim\frac{1}{24\zeta(2)}(\log L)^3.\]

Section 4 in the paper

Section 5

The conductor-separated saturated family

The proof of Theorem 1.1 uses a sparse subfamily of this fan. Write \(e=rs^2\), \(\rho(e)=rs\), with \(r,s\) squarefree but not sieved at small primes. Split the medium factor as \(d=d_-d_+\), where \(d_-\) is a small-prime part absorbed into a base residue class and \(d_+\) is rough.

We then choose moduli of the form \(Q=d_-d_+p\), where \(p\) is a large prime. The parameter ranges force \(\rho(e)>d\). This inequality gives the large-prime uniqueness input: two distinct compatible events cannot share their large prime \(p\).

Section 5 in the paper

Section 10

The base-conditioned CRT no-hit probability

If two events are compatible, the CRT gives \(\mathbb P(\mathcal A_i^{(b)}\cap\mathcal A_j^{(b)}) =1/\mathop{\rm lcm}(q_i,q_j)=(q_i,q_j)/(q_iq_j)\); if they are incompatible, the intersection is empty. Events with coprime moduli depend on disjoint coordinates.

Section 10 in the paper

Section 11

Transfer to finite intervals

Let \(K>0\) be a fixed sufficiently large constant and put \(R_b=\lfloor K\mu_b\rfloor\). Suppose that

\[P(z)X^{2R_b}\exp\{2\mu_b\}\le N\exp\{-4\mu_b\}.\]

Then, uniformly in \(b\),

\[\#\{n\le N:n\equiv b\pmod {P(z)},\ n\notin\mathcal A_i^{(b)} \text{ for every }i\in\mathcal I_b\} \le\frac{N}{P(z)}\exp\{-(1-o(1))\mu_b\}.\]

Section 11 in the paper

Section 12

Optimization, saturation lifting, and proof of the main theorem

The saturated mass law is \(\mu_b\asymp(\log X)^3\), uniformly in \(b\). Choose \(\log X=\alpha(\log N)^{1/4}\), where \(\alpha>0\) is a sufficiently small fixed constant. Then \(\mu_b\asymp_\alpha(\log N)^{3/4}\).

Complete proof of Theorem 1.1

Section 13

Applications of the saturated fan

The unconditional consequences include the geometric form in Corollary 2.5. We also record an exceptional-prime bound with the same exponential saving, a large-prime-factor propagation theorem, and a fixed-numerator version for the Erdős–Straus–Schinzel equation.

Prime denominators

There is an absolute constant \(c>0\) such that, for all \(N\ge3\),

\[\#\mathcal P_{\rm exc}(N)\ll N\exp\{-c(\log N)^{3/4}\}.\]

Fixed numerators

For each fixed integer \(m\ge2\), there are constants \(c_m,C_m>0\) such that, for all \(N\ge3\),

\[E_m(N)\le C_mN\exp\{-c_m(\log N)^{3/4}\}.\]

Geometric form

There is an absolute constant \(c>0\) such that, for all \(N\ge3\), for all but \(O\!\left(N\exp\{-c(\log N)^{3/4}\}\right)\) integers \(n\le N\), the log K3 surface \(U_n\) has an integral point in the positive chamber. Moreover, one may choose such a point on one of the rational fan curves \(C_{n,a}\).

Read the applications and proofs

Citation

@misc{kandukoori2026erdosstraus,
  title  = {An improved exceptional-set bound for the
            Erdős--Straus equation via saturated divisor fans},
  author = {Sushaan Kandukoori and Shrikrishna Ramesh
            and Pranava Kumar},
  year   = {2026},
  url    = {https://github.com/sushaan-k/erdos-straus-lean}
}

Formalization

Lean 4

A companion Lean 4 development formalizes the deduction of Theorem 1.1 from nine named theorems, either cited here or proved in the manuscript. It verifies the certificate identities, parameter inequalities, mass and tensorisation estimates, dependency bounds, finite transfer, optimization, and smooth lifting. The weighted reciprocal Bombieri–Vinogradov estimate and the long squarefree progression estimates are stated in the exact forms used here rather than derived from more primitive versions inside Lean; the repository records this boundary explicitly.

theorem erdos_straus_exceptional_set_bound : IsMainBound exceptionalCount

FORMALIZATION.md EscAnalytic/Paper.lean