A unified parametric approach to the Erdős--Straus conjecture with explicit solutions for a set of integers of natural density one

1. Problem Statement

The paper addresses the Erdős–Straus conjecture (1948), which posits that for every integer $n \ge 2$, there exist positive integers $x, y, z$ such that:
$$\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$
The author also considers the Sierpiński conjecture ($k=5$) and the generalized equation for any fixed integer $k \ge 4$:
$$\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$
While the conjecture has been verified numerically for $n$ up to $10^{18}$, a general proof remains elusive. The primary difficulty lies in the residue class $n \equiv 1 \pmod 4$ for the case $k=4$, which previous methods have struggled to cover comprehensively.

2. Methodology

The author develops a unified parametric approach based on a specific quadratic function derived from the Diophantine equation.

The Fundamental Function

The core of the methodology is the function $F_{x,t}^{(k)}(n)$, defined for fixed integers $k \ge 4$, $x \in \mathbb{N}^*$, and $t \in \mathbb{N}^*$:
$$F_{x,t}^{(k)}(n) = t^2(kx - n)^2 - 2nxt$$
The paper establishes a fundamental equivalence:
$$\frac{k}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \iff \exists x, t \in \mathbb{N}^* \text{ such that } F_{x,t}^{(k)}(n) = m^2 \text{ for some } m \in \mathbb{N}$$
If $F_{x,t}^{(k)}(n)$ is a perfect square $m^2$, explicit solutions are given by:
$$y = t(kx - n) + m, \quad z = t(kx - n) - m$$

Analysis of the Function $F$

The author rigorously analyzes the properties of $F_{x,t}^{(k)}(n)$ on its admissible domain $D_{x,t}^{(k)}$, defined by $n < kx$ and $t \ge \frac{2nx}{(kx-n)^2}$.

• Monotonicity: The function is strictly decreasing with respect to $n$ on this domain.
• Positivity: $F \ge 0$ on the domain.
• Zero Lemma: If $F(n_0) = 0$ for some $n_0$ in the domain, then $n_0$ is necessarily the upper bound (maximum element) of the admissible domain $D_{x,t}^{(k)}$. This implies that finding a solution where $F=0$ corresponds to finding the maximal $n$ for a given pair $(x,t)$.

3. Key Contributions and Results

A. Explicit Solutions for $n \not\equiv 1 \pmod 4$

For the classical case $k=4$, the author provides explicit symmetric solutions ($y=z$, implying $F=0$) for three of the four residue classes modulo 4:

1. $n \equiv 0 \pmod 4$: $n=4r \implies x=r+1, t=\frac{r(r+1)}{2}$.
2. $n \equiv 2 \pmod 4$: $n=4r+2 \implies x=r+1, t=(2r+1)(r+1)$.
3. $n \equiv 3 \pmod 4$: $n=4r+3 \implies x=r+1, t=2(4r+3)(r+1)$.

These constructions cover 75% of all integers, providing a complete proof for these classes using symmetric solutions.

B. Solutions for the Resistant Class $n \equiv 1 \pmod 4$

The most difficult case is $n \equiv 1 \pmod 4$. The author constructs a subfamily of solutions based on the existence of a specific divisor:

• Condition: Let $n = 4r + 1$. If $n$ possesses a divisor $b$ such that $b \equiv 3 \pmod 4$, an explicit solution exists.
• Construction: Set $x = \frac{n+b}{4}$. If $b|n$, then $t = \frac{n(n+b)}{2b^2}$ is an integer, and $F_{x,t}^{(4)}(n) = 0$, yielding the symmetric solution $y=z=tb$.
• Example: For $b=3$, this generates solutions for all $n \equiv 9 \pmod{12}$.

C. Natural Density Result

The paper proves that the set of integers $n \equiv 1 \pmod 4$ that do not possess a prime divisor $p \equiv 3 \pmod 4$ has natural density zero.

• Proof Strategy: The author defines set $B$ as integers $n \equiv 1 \pmod 4$ composed entirely of prime factors $p \equiv 1 \pmod 4$. Using Dirichlet series and Tauberian theorems, it is shown that the counting function $|B \cap [1, x]| = o(x)$.
• Implication: The set of integers $n \equiv 1 \pmod 4$ that do have a prime divisor $p \equiv 3 \pmod 4$ has natural density 1.

4. Significance and Conclusion

1. Verification for Density One: The Erdős–Straus conjecture is verified for a proportion of integers tending to 1. Specifically, it is proven for all $n \not\equiv 1 \pmod 4$ and for almost all $n \equiv 1 \pmod 4$ (those with a prime factor $\equiv 3 \pmod 4$).
2. New Families of Solutions: The work yields infinitely many new families of explicit symmetric solutions, including those not covered by Mordell's earlier results (e.g., specific residue classes modulo 840).
3. Structural Insight: The paper shifts the focus from purely numerical verification to the structural properties of the function $F_{x,t}^{(k)}(n)$. The "Zero Lemma" and the analysis of the admissible domain provide a new theoretical framework for attacking the conjecture.
4. Unified Framework: The approach successfully generalizes to the Sierpiński conjecture ($k=5$) and higher $k$, reducing the problem to finding parameters $(x,t)$ that make $F$ a perfect square.

In summary, Mballa provides a powerful parametric method that resolves the conjecture for 75% of integers explicitly and proves it holds for the remaining 25% with a set of exceptions of density zero, significantly advancing the theoretical understanding of the Erdős–Straus problem.