Solving Diophantine Equations

TL;DR

This paper discusses a method for partially solving Diophantine equations using empirical search within a defined domain, supported by computational tools, and reports on the partial solutions of 62 such equations.

Contribution

The authors introduce a practical approach for partial resolution of Diophantine equations using computational search and provide solutions for 62 equations, enhancing problem-solving tools.

Findings

Partial solutions for 62 Diophantine equations identified.

Development of computational tools in Mathcad for efficient search.

Method can be adapted to other programming languages and systems.

Abstract

In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\eta}({\pi}(x)) = {\pi}({\eta}(x)), where {\eta} is the Smarandache function and {\pi} is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful tools for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.