Parametric Algorithms for the 5-Modular Analog of ES (Sierpi\'nski): Structure of Solutions, Parameterization, and Constructive Proofs (SERP)

Abstract

We consider the problem of representing the fraction $5/P$ as a sum of three distinct unit fractions $1/A+1/B+1/C$ with $A<B<C$ and $A,B,C\in\mathbb{N}$. The case of primes $P\equiv 1 \pmod{5}$ is analyzed, where two constructive types of solutions arise: ED1 (exactly one denominator divisible by $P$, namely $C=cP$) and ED2 (exactly two denominators divisible by $P$, namely $B=bP$ and $C=cP$). Parametric constructions and enumeration algorithms are developed, including explicit transitions between ED1 and ED2. A deterministic algorithm is proposed, based on the intersection of a parametric lattice defined by pairs $(\alpha,d')$ with bounded boxes. For each fixed prime $P\equiv 1 \pmod{5}$ the algorithm constructively produces a solution. Using analytic methods such as the Bombieri--Vinogradov theorem and the Chebotarev density theorem, it is shown that the density of admissible parameters is high, which yields polylogarithmic search complexity in the average case. A strict complexity guarantee for all primes remains conditional and depends on the finite covering hypothesis. This study extends previous work for coefficient $4$ (the Erd\H{o}s--Straus conjecture) to coefficient $5$, transferring the same structure of parametrization and constructive solutions. Analytic applications provide averaging tools used for density estimates in parametric boxes.