Abstract:

In this paper we present methods of solving the Diophantine equations of the second degree. For each solution we determine the solvability conditions. Algorithms are presented in their general form. We illustrate their applications for specific examples.
This article was implemented for high school students with a focus on deeper, more intensive study of mathematics and computer science.
After mastering the methodology in the article, students demonstrated much better ability to solve these and similar tasks.
The author pays attention to the following goals in education for educators for further study of this issue.

Educational goal:
This contribution helps the teacher to orientate correctly in solving Diophantine equations, deepen, intensify his knowledge, expand horizons, and intensify thinking in the given field. Specifically
-it helps him analyze the problems correctly and in detail
- based on a detailed analysis, he will determine the right thinking strategy that directs him on
-synthesis of problem solving.
In the case of several possible solutions, it presents the possibility to compare the quality of the solutions, the difficulty of the solution and the quality, the problem solving efficiency.
Finally, it will enable the teacher to draw conclusions, compare, classify and decide on the best, most appropriate method for specific problems of the type.
All of these points of education point to the fact that this contribution in the main and deepest essence development of the teacher's creativity in teaching the pupil and also in the pupil creates a wide range of variable approaches, problem solving ideas in the next study.
This article is suitable as an additional material for teachers and students of secondary schools. Challenging tasks are suitable for students in mathematical classes.

Keywords:

Diophantine equation of the second degree, the Pythagorean equation, algorithm, condition, unknown, coefficient, the greatest common divisor.