arXiv subjects

Philemon Urbain Mballa

Publications and source records attributed to Philemon Urbain Mballa.

4 recordsLinked to original sources

arXiv2026

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

We develop 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 related generalizations. We introduce and analyze the properties of the fundamental function $F_{x,t}^{(k)}(n) = t^2(kx-n)^2 - 2nxt$, whose being a perfect square is equivalent to yielding a solution of these conjectures. In the classical Erdős--Straus case ($k=4$), for the residue classes $n \equiv 0,2,3 \pmod{4}$, we provide explicit symmetric solutions $y=z$, covering already 75\% of all integers. For the historically most resistant class $n \equiv 1 \pmod{4}$, we construct explicit symmetric solutions based on the existence of a divisor $b \equiv 3 \pmod{4}$, and we further 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. These results yield infinitely many new families of explicit solutions not covered by previous constructions, highlight the structural behavior of $F$.

math.NT↗

arXiv2026

Potential Relation Between the Riemann Zeta Function and the Polynomial Function $F$ of the Generalized Erdős--Straus Conjecture, Subject to its Analytic Continuation

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each $n\ge 1$ a decomposition $\frac{k}{n^s} = \frac{1}{x_s(n)}+\frac{1}{y_s(n)}+\frac{1}{z_s(n)}$ with $x_s(n), y_s(n), z_s(n) \in \mathbb{R}^*+$. Summing this equality over all integers brings forth the Riemann zeta function. Subject to an analytic continuation of the quantities $x_s(n), y_s(n), z_s(n)$ to complex values of $s$, one would obtain a new function $G_k(s)$ satisfying $G_k(s)=k\,ζ(s)$, thus establishing a deep connection between the structure of the conjecture and the zeros of $ζ$.

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arXiv2026

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

This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two new explicit analytical formulas. The first formula, which is a special case of the second, is based on a divisibility condition, whereas the second, more general formula, relies on the existence of a perfect square, which we conjecture to always hold. Under these conditions, the formulas verify the conjectures even for very large numerical values. Moreover, our method reduces the problem to the search for a suitable perfect square, thereby opening the way to a complete proof of these conjectures. In conclusion, we present open questions and conjectures to the mathematical community regarding the generalization of these formulas.

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arXiv2025

An Unexpected Connection Between the Discrete Zeta Function and the Erdos-Straus Conjecture Under Mballa's Conjecture

In this article, we establish an additive decomposition of the discrete zeta function (for $s \in \mathbb{N}^*$, $s > 1$), more precisely of the function $4(ζ(s)-1)$, as a series whose general term is of the form $1/x_n(s) + 1/y_n(s) + 1/z_n(s)$, where $x_n(s), y_n(s), z_n(s)$ are solutions of the Erdos--Straus conjecture under a personal conjecture (which I will refer to here as Mballa's Conjecture) that I formulated by parametrization in the article: arXiv:2502.20935. This connection thus builds a bridge between analysis and Egyptian fractions in general, and the Erdos--Straus conjecture in particular.

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