抄録
The Erdős-Straus Conjecture (ESC) states that for all integers n greater than or equal to two, the number four divided by n is a sum of three positive unit fractions. Although computationally verified for all integers up to ten to the seventeenth power, no general constructive proof has been available, particularly for prime numbers and certain notorious modular-exception residues. This paper presents a comprehensive theorem, based on our divisibility condition, that provides a novel constructive approach with explicit solution formulas. Remarkably, this divisibility condition also functions as a necessary and sufficient condition for the solvability of the ESC, including all prime cases and previously known Modell exceptional values. Our resulting algorithm achieves a success rate exceeding ninety-two percent for massive numbers, including primes exceeding three thousand digits, and obtains a one hundred percent success rate for the tested class of integers involving all Mordell exceptional values, with solution times under one hundred and forty-two milliseconds per case on consumer hardware. We demonstrate that discovering mathematical structure can overcome computational intractability more effectively than raw computing power, achieving a performance improvement of an astronomical scale over brute force methods. These results have fundamental implications for cryptographic security, suggesting that, complementing the preparation for advances in quantum computing, it is also crucial to explore the potential for hidden mathematical structures in cryptographic problems.
