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On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$

Joel Louwsma

TL;DR

This work addresses representing $1$ as a sum of unit fractions with restricted denominators $x_i \in S(k)=\{2^a k^b : a\in\{0,1,2\}, b\ge0\}$ for $k$ not a power of $2$. It introduces a count-based array encoding $c_{a,b}$ and a set of reduction/expansion moves that manipulate these arrays to generate all such solutions, with reductions guaranteed to decrease the term count. The main contributions are a complete algorithmic framework to enumerate all solutions for given $k$ and $n$, and a precise criterion determining when nontrivial solutions exist, depending on $k$ modulo 4, plus explicit constructive cases. Computational results from a Python implementation illustrate counts for small $k$ and $n$, and the method yields a practical tool for exploring structured Egyptian fraction decompositions and their combinatorial structure.

Abstract

We give an algorithm that produces all solutions of the equation $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$, where $k$ is a fixed positive integer that is not a power of $2$, $a$ is an element of $\{0,1,2\}$ that can vary from term to term, and $b$ is a nonnegative integer that can vary from term to term. We also completely characterize the pairs $(k,n)$ for which this equation has a nontrivial solution in integers of this form.

On solutions of $\sum_{i=1}^n 1/x_i = 1$ in integers of the form $2^a k^b$

TL;DR

This work addresses representing as a sum of unit fractions with restricted denominators for not a power of . It introduces a count-based array encoding and a set of reduction/expansion moves that manipulate these arrays to generate all such solutions, with reductions guaranteed to decrease the term count. The main contributions are a complete algorithmic framework to enumerate all solutions for given and , and a precise criterion determining when nontrivial solutions exist, depending on modulo 4, plus explicit constructive cases. Computational results from a Python implementation illustrate counts for small and , and the method yields a practical tool for exploring structured Egyptian fraction decompositions and their combinatorial structure.

Abstract

We give an algorithm that produces all solutions of the equation in integers of the form , where is a fixed positive integer that is not a power of , is an element of that can vary from term to term, and is a nonnegative integer that can vary from term to term. We also completely characterize the pairs for which this equation has a nontrivial solution in integers of this form.
Paper Structure (6 sections, 9 theorems, 49 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 6 sections, 9 theorems, 49 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The moves
  3. The algorithm
  4. When there exist nontrivial solutions
  5. Solutions for k=3 and small n
  6. Solutions for k=5 and small n

Key Result

Lemma 1

Let $k$ be a positive integer that is not a power of $2$, let $A$ be an array corresponding to a solution of eq:main with each $x_i\in S(k)$, and let $\beta$ be the index of the top nonzero row of $A$.

Figures (2)

  • Figure A.1: Arrays corresponding to solutions of \ref{['eq:main']} with $n\leq4$ and each $x_i\in S(3)$. The labels on the edges are the numbers of the moves used. The trivial solutions are shown in red.
  • Figure B.1: Arrays corresponding to solutions of \ref{['eq:main']} with $n\leq5$ and each $x_i\in S(5)$. The labels on the edges are the numbers of the moves used. The trivial solutions are shown in red.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Example 5
  • Theorem 6
  • Proposition 7
  • ...and 9 more