Scientific Bulletin. Physical and Mathematical Research


This article is based on the lectures for gifted students of the faculty of Physics and Mathematics on the solution of Diophantine equations in science circles.

If the number of unknowns involved in a system of equations exceeds the number of equations, such equations are called Diophantine equations or indeterminate equations. Specifically, equations of the form

3x-5y=8, x2+3xy-y2=12,

x3+y2-3x+5=0, x3+y3=z3,… are indefinite equations.

Many of the equation or system of equations determine all the numbers to find solutions to the most common examples. Short multiplication formulas, theory serve as the primary means of logical thinking in the solution of mathematical equation.

However, in the theory of division of equations of the form such as ax+by=c, there are formulas for solving the corresponding fractions. Firstly, we presented the basic properties of solving equations of this type in integer numbers based on the theory of division, formulas for solving them, and finally described the examples of solving them by appropriate fractions. In the next step, we give examples of solving high-level indeterminate equations.

The importance of studying the solution of linear and high-level indeterminate equations in whole numbers is obvious in solving many practical problems in such areas as finance, economics, technology. Therefore, these issues are included in the programs of entrance exams to higher education institutions. We believe that theoretical information in this article and as well as specific examples with solutions can be used by applicants while preparing for entrance exams to higher education institutions and their teachers, as well as teachers mathematics who organize extracurricular activities in secondary schools. For this purpose, at the end of the article there are a number of independent works that can be solved on the basis of examples and theoretical data, as well as to strengthen knowledge on these issues. The set of examples and questions provided by the State Test Center under the Cabinet of ministers of the Republic of Uzbekistan, intended for entrance exams to higher education institutions in different years is was widely being used in covering the topic of the article.

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