Abstract:

This paper is focused on a special type of Diophantine equations similar to the Pythagorean theorem, i.e. solving the equation x²+y²=α·z², where x,y,z ∈ Z are variables for different integer values of α. It is suitable for students of mathematics and informatics as well as for teachers of the mentioned scientific fields as work material. In this paper, new techniques and solution strategies are presented, as well as effective algorithms for solving tasks associated with them.
Educational objective: The mentioned contribution helps the teacher to properly orient himself in the solution of Diophantine equations of the mentioned types, to deepen and intensify his knowledge, to broaden horizons, to intensify thinking in the given area. These techniques help students correctly analyze problems, choose an appropriate solution strategy, understand more easily the depth of the difficulty of the task, and construct effective solution algorithms.
Individual task solving techniques were tested on high school students as well as on university students. After studying the mentioned techniques, the students demonstrated excellent results in solving specific tasks. Particularly talented students were able to solve even very difficult tasks that they had not been able to handle before.

Keywords:

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