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

TL;DR

This paper develops parametric algorithms and constructive proofs for representing 5/P as a sum of three unit fractions, extending methods from the Erdős–Straus conjecture for coefficient 4, with complexity analysis based on advanced number theory.

Contribution

It introduces new parametric constructions, enumeration algorithms, and a deterministic solution method for the 5-modular case, extending previous work on the coefficient 4 case.

Findings

High density of admissible parameters for primes congruent to 1 mod 5

Polylogarithmic average-case search complexity

Conditional strict complexity guarantees based on finite covering hypothesis

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.