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Exceptions to the Erd\H os--Straus--Schinzel conjecture

Carl Pomerance, Andreas Weingartner

TL;DR

This work analyzes the Erdős–Straus–Schinzel problem of expressing m/n as a sum of three unit fractions. By combining Type I/II structural decompositions with Brun–Titchmarsh, large-sieve methods, and extensive computational data, it proves that any bound $n_m$ must satisfy $n_m\ge \exp(m^{1/3+o(1)})$ and establishes a numerically explicit version with $m\ge 6.52\times10^9$ giving a prime in $(m^2,2m^2)$ for which $m/p$ is not representable; it also derives an upper bound on the exceptional set via Vaughan’s large-sieve approach and extends the investigation to sums of $j$ unit fractions. The results provide a clear transition between typically-false and typically-true regimes as $n$ grows, and they include detailed numerical evidence and explicit bounds across many gcd-case analyses, culminating in a rigorous explicit theorem with broad generalizations and empirical validation.

Abstract

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpiński, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge19$. In addition we generalize a result of Vaughan to show that for each $m$, most $n$'s have $m/n$ representable, and we prove a result generalizing the problem to the sum of $j$ unit fractions.

Exceptions to the Erd\H os--Straus--Schinzel conjecture

TL;DR

This work analyzes the Erdős–Straus–Schinzel problem of expressing m/n as a sum of three unit fractions. By combining Type I/II structural decompositions with Brun–Titchmarsh, large-sieve methods, and extensive computational data, it proves that any bound must satisfy and establishes a numerically explicit version with giving a prime in for which is not representable; it also derives an upper bound on the exceptional set via Vaughan’s large-sieve approach and extends the investigation to sums of unit fractions. The results provide a clear transition between typically-false and typically-true regimes as grows, and they include detailed numerical evidence and explicit bounds across many gcd-case analyses, culminating in a rigorous explicit theorem with broad generalizations and empirical validation.

Abstract

A famous conjecture of Erd\H os and Straus is that for every integer , can be represented as , where are positive integers. This conjecture was generalized to by Sierpiński, and then Schinzel conjectured that for every integer there is a bound such that the fraction is the sum of 3 unit fractions for all integers . Leveraging and generalizing work of Elsholtz and Tao, we show that if exists it must be at least ; that is, there are numbers this large for which is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for there is a prime with not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound . In addition we generalize a result of Vaughan to show that for each , most 's have representable, and we prove a result generalizing the problem to the sum of unit fractions.

Paper Structure

This paper contains 14 sections, 20 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. Some basic thoughts
  3. A lower bound for the number of exceptional primes
  4. An upper bound
  5. The general case
  6. Empirical data
  7. Numerically explicit estimates: primes with Type I representations
  8. The case $\gcd(f,6)=1$.
  9. The case $\gcd(f,6)=2$.
  10. The case $\gcd(f,6)=3$.
  11. The case $(f,6)=6$.
  12. Numerically explicit estimates: primes with Type II representations
  13. Proof of Theorem \ref{['th-explicit']}
  14. Some final thoughts

Key Result

Theorem 1.1

For each $\epsilon>0$ there is a bound $m(\epsilon)$ such that for each $m\ge m(\epsilon)$ there is some $n>\exp(m^{1/3-\epsilon})$ with $m/n$ not the sum of $3$ unit fractions.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 27 more