What question did this study set out to answer?

To investigate the Erdős-Straus conjecture on Egyptian fractions (4/n = 1/x + 1/y + 1/z) and establish explicit algebraic solutions across modular families alongside formal machine-checked proofs.

August 20, 20260 citationsOpen Access

On the Erdős-Straus Conjecture on Egyptian Fractions

Ver artículo completo

CNCharles EDOU NZE

Puntos clave

To investigate the Erdős-Straus conjecture on Egyptian fractions (4/n = 1/x + 1/y + 1/z) and establish explicit algebraic solutions across modular families alongside formal machine-checked proofs.
Derived a universal 3-parameter polynomial identity and proved a composite reduction theorem to reduce the global conjecture to prime values of n.
Performed exhaustive combinatorial classifications across congruence classes and fully verified the proofs using the Lean 4 interactive theorem prover.
Demonstrated exact integer solutions for 23 out of 24 residue classes modulo 24 (n ≢ 1 mod 24), resolving the conjecture for at least 95.833% of all natural numbers.
Provided complete parametric derivations for primary congruence classes, including n ≡ 0 (mod 2), n ≡ 0 (mod 3), n ≡ 3 (mod 4), n ≡ 2 (mod 3), and n ≡ 5 (mod 8), accompanied by 100% Lean 4 formal verification.

Resumen

This preprint provides an exhaustive mathematical treatise on the Erdős-Straus Conjecture (Problem #108 in Paul Erdős' collection), formulated by Paul Erdős and Ernst G. Straus in 1948. The conjecture asserts that for every integer n 2, the Egyptian fraction Diophantine equation: 4n = 1x + 1y + 1z admits a solution in positive integers (x, y, z) (N>₀) ³, or equivalently in polynomial form 4xyz = n (xy + yz + xz). Key Mathematical Results Vaughan, 1970). 100% Machine-Checked Verification in Lean 4 Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub: https: //github. com/flouzzy/erdos-problems Primary MSC (2020): 11D68, 11A07, 68V20, 11Y50, 11N36. Keywords: Erdős-Straus Conjecture, Egyptian Fractions, Diophantine Equations, Modular Reduction, Sieve Methods, Formal Verification, Lean 4, Mathlib.

Preguntar a la IA

Me gusta

Guardar

Ver artículo completo

Preguntar a la IA

Me gusta

Guardar

Ver artículo completo

On the Erdős-Straus Conjecture on Egyptian Fractions

Puntos clave

Resumen

Cite This Study