Parametric Algorithms for the 5-Modular Analog of ES (Sierpiński): Structure of Solutions, Parameterization, and Constructive Proofs (SERP)
E. Dyachenko
TL;DR
This work extends the Erdős–Straus framework to the 5-modular Sierpiński problem by constructing two parametric solution families, ED1 and ED2, for primes $P\equiv 1\pmod{5}$. It preserves the lattice-box/covering approach from the coefficient-4 case while updating core identities to accommodate the coefficient 5, enabling explicit, factorization-free constructions with polylogarithmic average-case search complexity. ED1 uses the kernel $(\gamma A-c)(\gamma B-c)=c^2$ with $5c-1=\gamma P$, while ED2 relies on the kernel $(5b-1)(5c-1)=5P\delta+1$ and a normalization to a linear-affine lattice; transitions between ED1 and ED2 (convolution/anticonvolution) ensure completeness and reduce multiplicity. Analytic tools such as Bombieri–Vinogradov and Chebotarev density underpin density estimates in parametric boxes, yielding practical algorithms and averaging results, with conditional guarantees tied to finite covering hypotheses; concrete examples and experiments illustrate the approach and its potential for broader generalizations.
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 $(α,d')$ 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 Erdős--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.