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Solutions to $\sum_{i=1}^n 1/x_i=1$ in integers $p^a\,q^b$ with $p$ and $q$ two set primes

Claire I. Levaillant

TL;DR

The paper studies representations of the unit as a sum of $n$ unit fractions with denominators $x_i$ of the form $p^a q^b$, where $p$ and $q$ are distinct primes and each appears at least once. It develops a structured, algorithmic framework based on a two-dimensional $p$–$q$ exponent table, admissibility criteria for the top row, and push/left/right moves to enumerate all such solutions of length $n$ with bounded $p$-valuation, including the special case $p=2$. Key results include a fundamental admissibility bound $q < p^{eta} n$, a constructive existence result for $p=2$ with large $n$, and corollaries that bound the admissible primes $q$ for fixed $n$ and width; finally, the algorithm is fully described to generate all solutions while controlling for duplicates. These contributions extend prior work by handling arbitrary two-prime denominators with unbounded exponents and provide a practical enumeration method for Egyptian-fraction decompositions under the stated restrictions.

Abstract

We present an algorithm for computing all the solutions in not necessarily distinct integers to the decomposition of the unit into a sum of unit fractions with denominators $p^a.q^b$ where $p$ and $q$ are two distinct primes, each appearing at least once in the solution.

Solutions to $\sum_{i=1}^n 1/x_i=1$ in integers $p^a\,q^b$ with $p$ and $q$ two set primes

TL;DR

The paper studies representations of the unit as a sum of unit fractions with denominators of the form , where and are distinct primes and each appears at least once. It develops a structured, algorithmic framework based on a two-dimensional – exponent table, admissibility criteria for the top row, and push/left/right moves to enumerate all such solutions of length with bounded -valuation, including the special case . Key results include a fundamental admissibility bound , a constructive existence result for with large , and corollaries that bound the admissible primes for fixed and width; finally, the algorithm is fully described to generate all solutions while controlling for duplicates. These contributions extend prior work by handling arbitrary two-prime denominators with unbounded exponents and provide a practical enumeration method for Egyptian-fraction decompositions under the stated restrictions.

Abstract

We present an algorithm for computing all the solutions in not necessarily distinct integers to the decomposition of the unit into a sum of unit fractions with denominators where and are two distinct primes, each appearing at least once in the solution.
Paper Structure (3 sections, 8 theorems, 63 equations)

This paper contains 3 sections, 8 theorems, 63 equations.

Table of Contents

  1. Introduction and preliminary results
  2. Repartition of the primes
  3. Description of the algorithm

Key Result

Lemma 1

If $k_1\,\,k_2\,\dotsm\, k_{\alpha_p+1}$ is the top row of a solution in $p^aq^b$ with $b\geq 0$ and $0\leq a\leq \alpha_p$, then $q$ divides $\sum_{i=1}^{\alpha_p+1}k_ip^{\alpha_p+1-i}$.

Theorems & Definitions (21)

  • Lemma 1: Admissibility of a top row
  • proof
  • Definition 2: Push value
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Definition 8
  • Proposition 9
  • proof
  • ...and 11 more