Partial Resolution of the Erdös-Straus, Sierpinski, and Generalized Erdös-Straus Conjectures Using New Analytical Formulas

1. Problem Statement

The paper addresses three major open problems in number theory concerning Egyptian fractions:

1. The Erdős–Straus Conjecture (1948): For every integer $n \ge 2$, there exist positive integers $x, y, z$ such that:
   $$\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$
2. The Sierpiński Conjecture: A variation where the numerator is 5:
   $$\frac{5}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$
3. The Generalized Conjecture: For any integer $a \ge 5$, there exists a threshold $N_0$ such that for all $n \ge N_0$, there exist $x, y, z \in \mathbb{N}^*$ satisfying:
   $$\frac{a}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$$

While the conjectures have been verified numerically up to $n = 10^{17}$ and partially proven by Mordell (who showed validity for all $n$ not congruent to specific residues modulo 840), a general analytical proof or explicit formula generating solutions for all $n$ remains elusive.

2. Methodology

The author proposes a unified analytical approach based on the structural analysis of the underlying Diophantine equation and parametric substitution.

Core Derivation Steps:

1. Transformation: The equation $\frac{a}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is rewritten as:
   $$\frac{ax - n}{nx} = \frac{y+z}{yz}$$
   This implies the condition $ax - n > 0$, leading to the constraint $x \ge \lfloor \frac{n}{a} \rfloor + 1$.

2. Parameterization: The author introduces a parameter $t \in \mathbb{Q}^*_+$ to express the sum and product of $y$ and $z$:
   $$y + z = (ax - n)(2t)$$
   $$yz = nx(2t)$$
   This transforms the problem into finding roots of a quadratic equation in $V$:
   $$V^2 - (ax - n)(2t)V + 2nxt = 0$$

3. Discriminant Analysis: For $y$ and $z$ to be integers, the discriminant $\Delta$ of the quadratic equation must be a perfect square:
   $$\Delta = 4t^2(ax - n)^2 - 8nxt = 4[t^2(ax - n)^2 - 2nxt]$$

3. Key Contributions: Two New Analytical Formulas

The paper derives two distinct formulas based on conditions imposed on the discriminant.

Formula 1: The Divisibility Condition

The author sets a specific condition where the term inside the discriminant simplifies to a square by choosing $t$ such that the discriminant becomes a perfect square of a rational number.

• Derivation: By setting $t = k$ where $k = \frac{2nx}{(ax-n)^2 - 1}$, the discriminant becomes $\Delta = (2k)^2$.
• Resulting Solutions:
  $$y = \frac{2nx}{ax - n + 1}, \quad z = \frac{2nx}{ax - n - 1}$$
• Condition: This formula provides a valid solution if and only if both $(ax - n + 1)$ and $(ax - n - 1)$ divide $2nx$.
• Significance: This reduces the problem to a search for an integer $x$ satisfying specific divisibility constraints.

Formula 2: The Perfect Square Condition (General Case)

This is presented as the more fundamental approach. It posits that a solution exists if the expression inside the discriminant is a perfect square.

• Condition: There exist integers $x, t$ (or rational $t$) such that:
  $$t^2(ax - n)^2 - 2nxt = q^2$$
  where $q \in \mathbb{N}$ (or $\mathbb{Q}^+$).
• Resulting Solutions:
  $$y = t(ax - n) - q, \quad z = t(ax - n) + q$$
• Significance: This formula subsumes the first one. The author conjectures that for every $n$, such a pair $(x, t)$ always exists.

4. Results and Numerical Verification

The paper provides extensive numerical evidence supporting the validity of these formulas:

• Mordell's Exceptional Cases: The author tested the Second Formula against the specific values of $n$ (e.g., $121, 169, 841, \dots$) that Mordell's modular arithmetic approach could not resolve.
  • Range [2, 5000]: 100% success rate (35 exceptional cases found and solved).
  • Range [5000, 10000]: 100% success rate (36 exceptional cases found and solved).
• Generalization: The method was successfully applied to:
  • The Sierpiński conjecture ($a=5$).
  • Generalized cases ($a=6, 11$).
• Tables: The paper includes detailed tables (Tables 1–9) listing specific values of $n, x, y, z$ (and $t, q$ where applicable) that satisfy the equations, verifying the identity $\frac{a}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ with high precision.

5. Significance and Open Questions

Significance

• Unified Framework: The paper offers a single analytical framework that handles even numbers, odd numbers, primes, and generalized numerators ($a \ge 4$).
• Reduction of Complexity: It reduces the infinite search space of the Diophantine equation to the search for a suitable perfect square or a specific divisibility condition.
• Pedagogical Value: The method provides a clear, constructive way to find solutions without relying on brute-force enumeration.

Open Questions and Conjectures

The author explicitly states that while the formulas work numerically, a rigorous proof that the conditions are always met for all $n$ is required to fully resolve the conjectures. The paper poses the following challenges to the mathematical community:

1. Explicit Construction: Can an explicit formula for $x_0$ (and $t_0$) be derived as a function of $n$ that guarantees the divisibility or perfect square condition?
2. Existence Proof: Can the existence of such $x$ and $t$ be proven via induction or other number-theoretic methods?
3. Conjectures Proposed:
   • Conjecture 1 (Strong): For every $n \ge 2$, there exist integers $x, t$ such that $t^2(4x-n)^2 - 2nxt$ is a perfect square.
   • Conjecture 4 (Generalized): For every $a \ge 5$, there exists a threshold $N_1$ such that for all $n \ge N_1$, integer solutions for the perfect square condition exist.

Conclusion

Philemon Urbain Mballa's paper does not claim a complete proof of the Erdős–Straus conjecture but provides a partial resolution by establishing two powerful analytical formulas. These formulas successfully generate solutions for all tested cases, including the historically difficult "exceptional" cases left by Mordell. The work shifts the focus of the problem from finding solutions directly to proving the existence of specific integer parameters ($x, t$) that satisfy a perfect square condition, offering a promising new pathway for a complete proof.