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 $\dfrac{5}{P}$ as a sum of three distinct unit fractions with natural denominators: $$ \frac{5}{P} \;=\; \frac{1}{A} + \frac{1}{B} + \frac{1}{C}, \qquad A < B < C,\quad A,B,C\in\mathbb{N}. $$ We analyze the case of primes $P \equiv 1 \pmod{5}$, for which two types of solutions are distinguished: $\EDone$ (exactly one denominator is divisible by $P$, namely $C=cP$) and $\EDtwo$ (exactly two denominators are divisible by $P$, namely $B=bP$ and $C=cP$). Parametric constructions and search algorithms are developed, including constructive transitions between the types of solutions. The study is a continuation of the work for coefficient 4 (the Erdős–Straus conjecture) Constructive Proofs of the Erdős–Straus Conjecture for Prime Numbers of the Form P ≡ 1 (mod 4)}; here, a similar structure of parameterization and solutions is transferred to coefficient 5. (Sierpiński Variant) In analytic applications, averaging tools (Bombieri–Vinogradov, the larger sieve, Chebotarev) are presented, used for density estimates in parametric boxes.