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Properties of the Function \(F_{x,t}^{(k)}(n)\) with Applications to the Erdős--Straus, Sierpiński Conjectures and Their Generalizations

Philemon Urbain Mballa

Abstract

This article develops a parametric approach to study the Diophantine equation \(\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\), underlying the Erdős--Straus (\(k=4\)), Sierpiński (\(k=5\)), and their generalizations. We introduce and analyze the fundamental function \(F_{x,t}^{(k)}(n) = t^2(kx-n)^2 - 2nxt\), whose perfect square values are equivalent to solutions of the conjectures. For any fixed pair \((x,t)\), we define its admissible domain \(\mathcal{D}_{x,t}^{(k)}\) and prove that on this domain, \(F\) is strictly decreasing, non-negative, and converges to its minimum. A key result is the Zero Lemma: if \(F(n_0)=0\) for some \(n_0\) in the domain, then \(n_0\) is necessarily the upper bound of \(\mathcal{D}_{x,t}^{(k)}\), and such zeros of \(F\) yield explicit symmetric solutions with \(y=z\). As an illustration, in the classical Erdős--Straus case (\(k=4\)), we explicitly construct symmetric solutions \(y = z\) for all integers \(n \equiv 0,2,3 \pmod{4}\), covering already \(75\%\) of all integers. For the remaining class \(n \equiv 1 \pmod{4}\), which is traditionally more challenging, we construct explicit symmetric solutions based on the existence of a divisor \(b \equiv 3 \pmod{4}\), and we show that this condition is satisfied for almost all such integers: the set of exceptions has natural density zero. Consequently, the Erdős--Straus conjecture is verified for a proportion of integers tending to \(1\) in this class. In particular, we obtain infinitely many new explicit families of symmetric solutions for numbers not covered by Mordell's theorem. These results elucidate the structural behavior of \(F\) and provide a unified framework for generating large families of solutions.

Properties of the Function \(F_{x,t}^{(k)}(n)\) with Applications to the Erdős--Straus, Sierpiński Conjectures and Their Generalizations

Abstract

This article develops a parametric approach to study the Diophantine equation , underlying the Erdős--Straus (), Sierpiński (), and their generalizations. We introduce and analyze the fundamental function \(F_{x,t}^{(k)}(n) = t^2(kx-n)^2 - 2nxt\), whose perfect square values are equivalent to solutions of the conjectures. For any fixed pair \((x,t)\), we define its admissible domain \(\mathcal{D}_{x,t}^{(k)}\) and prove that on this domain, is strictly decreasing, non-negative, and converges to its minimum. A key result is the Zero Lemma: if \(F(n_0)=0\) for some in the domain, then is necessarily the upper bound of \(\mathcal{D}_{x,t}^{(k)}\), and such zeros of yield explicit symmetric solutions with . As an illustration, in the classical Erdős--Straus case (), we explicitly construct symmetric solutions for all integers , covering already of all integers. For the remaining class , which is traditionally more challenging, we construct explicit symmetric solutions based on the existence of a divisor , and we show that this condition is satisfied for almost all such integers: the set of exceptions has natural density zero. Consequently, the Erdős--Straus conjecture is verified for a proportion of integers tending to in this class. In particular, we obtain infinitely many new explicit families of symmetric solutions for numbers not covered by Mordell's theorem. These results elucidate the structural behavior of and provide a unified framework for generating large families of solutions.
Paper Structure (1 section, 4 theorems, 85 equations, 1 table)

This paper contains 1 section, 4 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction

Key Result

Proposition 1

Let $k \ge 4$ and $(x,t) \in \mathbb{N}^{*2}$ be fixed. For every $n \in \mathcal{D}_{x,t}^{(k)}$, we have: Moreover, for all $n_1, n_2 \in \mathcal{D}_{x,t}^{(k)}$ with $n_1 > n_2$, we have: Thus, $n \mapsto F_{x,t}^{(k)}(n)$ is strictly decreasing on $\mathcal{D}_{x,t}^{(k)}$.

Theorems & Definitions (10)

  • Proposition 1: Decrease and positivity on the domain
  • proof
  • Proposition 2: Convergence and boundedness on the domain
  • proof
  • proof
  • proof
  • Proposition 3: Explicit symmetric solutions for three residue classes
  • proof
  • Proposition 4: Explicit symmetric solutions for a subfamily of $n \equiv 1 \pmod{4}$
  • proof