Abstract:

The article focuses on the specific Diophantine equations, their solution, and the possibility of implementing them in mathematics - competitions, and lessons. High school students should be able to solve various types of Diophantine equations. The article focuses on the particular type of Diophantine equations, their solution, and their introduction to the educational process. This paper also provides the solution to a generalized type of exponential Diophantine equation, including the proof in which we use elementary mathematics methods.

Let us mention that the Diophantine equations play an important role in mathematical analysis, especially equation solving. The Diophantine equations are not standard problems in teaching mathematics in high school. However, the current trend is to introduce these non-standard problems into teaching and allow students to explore the solution themselves. Even though the Diophantine equation can be solved using only methods of high school mathematics, during the educational process, it turns out that many students have no idea how to solve problems of the given type.

In this contribution, we focus on a specific type of Diophantine equation and solve it using only elementary mathematics. We demonstrate the methods of solution in various mathematical tasks. The paper can be helpful to mathematics teachers who might use it as material in preparing gifted and talented students for mathematical competitions.

We would also like to point out the possibility of integrating these problems into the learning process using the help of ICT. The described solution methods can be easily transformed into education in secondary schools. The results can be effortlessly verified within ICT, for example, using the Wolfram Mathematica, or directly visualized using GeoGebra.

Keywords:

Education, Diophantine, equation, exponential, elementary mathematics.