EQUATIONS IN INTEGERS IN OLIMPIC MATHEMATICS

Abstract

Formulation of the problem. The base content of math courses of institutions of general secondary education does not intend acquaintance with the theory of equations in integers or equations of Diophantus. Simultaneously we often meet such equations among the problems of student’s math competitions of different level. The work is devoted to carrying out analysis of types of problems on the theme “Equations in integers” at the up-to-date math competitions of students of institutions of general secondary education and basis methods of solving such problems.

Materials and methods. In the process of carried out investigation such theoretical methods as studying and analyzing the determined sources of information, providing reasoning of deductive and inductive character together with such practical methods as solving different problems and tasks of the determined theme, working out the detail explanations to the received solutions are used.

Results. The role and the place of Diophantine equations in the content of problems of schoolchildren’s math competitions beginning from the level of the III stage of the All-Ukrainian Math Olympiad are elucidated. Such types of problems as the tasks on solving linear Diophantine equations with different numbers of variables, some tasks connected with an equation of Pell, the tasks on establishing some properties of solutions of the given Diophantine equation, tasks on solving systems of Diophantine equations, tasks on finding some sequences of integer numbers are marked out. Among the most spread methods that are used in school Olympic mathematics for solving the problems such methods as canonical decomposition of natural numbers, decomposition of the one or the both parts of the equation onto multipliers, the method of mathematical induction and the method of estimation are distinguished. Examples of solutions to the corresponding problems are given.

Conclusions. Inclusion acquaintance with the beginnings of the theory of Diophantine equations to the content of math courses of institutions of general secondary education is undoubtedly useful for the all-round development of students, first of all from the position of forming and developing skills in their logical thinking. The represented analysis of the types and methods of solution of the corresponding problems can be useful to working teachers for organizing the work of math circles and conducting the elective courses according to the themes that are connected with equations in integers. The creation of detail synopses of such courses needs further working out.