Generalized Erdős-Straus and Schinzel-Sierpiński Conjectures

This preprint provides an exhaustive mathematical treatise on the Schinzel-Sierpiński Conjecture on Egyptian Fractions (Problem #26 in Paul Erdős' collection, 1956), which extends the classical Erdős-Straus conjecture (a = 4) to arbitrary numerators a ≥ 1, focusing specifically on Sierpiński's conjecture for 5/n. The conjecture asserts that for every fixed integer a ≥ 1, there exists an integer threshold Na such that for all integers n ≥ Na, the fraction a/n can be expressed as the sum of three positive Egyptian unit fractions: 5 / n = 1 / x + 1 / y + 1 / z, &qquad; x, y, z ∈ ℕ≥1 This treatise develops the universal modular identity families covering residue classes modulo 5, modulo 4, modulo 20, and presents complete computer-certified solutions for all prime denominators p ≤ 100 alongside the Elsholtz-Tao (2014) analytic framework on representation counting functions. Key Mathematical Results & Contributions Global Prime Reduction: Rigorous proof that if n = p · m is composite with m ≥ 2, any 3-term Egyptian decomposition for 5/p immediately scales to a valid decomposition for 5/n, reducing the conjecture strictly to prime denominators. Universal Parametric Modular Families: Complete algebraic derivations of polynomial triples (x(k), y(k), z(k)) covering prime residues p ≡ 2, 3, 4 &pmod; 5. Explicit Verified Solutions for Small Primes: Exhaustive enumeration and verification of positive integer solutions (x, y, z) for all primes p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}. Elsholtz-Tao Analytic Bounds (2014): Overview of the Bombieri-Vinogradov sieve methods establishing that exceptional prime denominators have asymptotic density zero. 100% Machine-Checked Verification in Lean 4: Rational fraction identities, 5/n field simplifications, and explicit parametric solutions for modulo classes are formally certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/erdos-problemsInteractive Portal: https://maths-proofs.edounze.com/#erdos-26-schinzel-sierpinski Primary MSC (2020): 11D68, 11A07, 11N36, 68V20, 11P83.Keywords: Erdős-Straus Conjecture, Schinzel-Sierpiński Conjecture, Egyptian Fractions, Diophantine Equations, Modular Arithmetic, Elsholtz-Tao Theorem, Formal Verification, Lean 4, Mathlib.