Based on the Fibonacci-Sylvester algorithm, we introduce a new elementary method in terms of the Fibonacci-Sylvester algorithm for studying the Erdős-Straus conjecture (ESC), that is, the case of all positive integer solutions of the Diophantine equation . Here, only elementary methods are used to provide the general solution expressions for its all positive integer solutions. Using this new method, we provide a new proof of the Mordell theorem, which states that  has a expression as the sum of three unit fractions for every natural number n, except possibly for the numbers of the form  with In addition, we also explore the situation of the Erdős-Straus conjecture when n(mod 1320) and n(mod 9240), and give the corresponding general solutions for all positive integer solutions of the equation, which is helpful for solving the problem of ESC and promoting the related research.

Egyptian fraction, Erdős-Strauss conjecture, Fibonacci-Sylvester algorithm, integral solution.

Received: April 20, 2024; Revised: July 15, 2024; Accepted: August 10, 2024; Published: August 24, 2024

How to cite this article: Xiaodan Yuan, On the Diophantine equation , JP Journal of Algebra, Number Theory and Applications 63(5) (2024), 459-480.

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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