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

We consider the problem of representing the fraction $5\mathrm{/}P$ as a sum of three distinct unit fractions $1\mathrm{/}A+1\mathrm{/}B+1\mathrm{/}C$ with $A<B<C$ and $A,B,C\in \mathbb{N}$. The case of primes $P\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$ 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 $(\alpha ,d^{\mathrm{\prime }})$ with bounded boxes. For each fixed prime $P\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$ 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.