A generalization of Diophantus' substitutions

The study considers the Diophantus' methods of solving certain systems of nonlinear equations given in the "Arithmetica" treatise, and further suggests the generalization of solutions founded by Diophantus.

The author regards the method of generalization of mathematical theories and statements as a means of developing students' creative thinking. It formates their readiness to develop creative thinking in their future pupils.

To archive this goal, the work proposes a historical approach based on using some facts from the history of mathematics, as well as famous mathematical problems. The latter include the problems from ancient treatises and the problems, formulated by famous mathematicians.

The study investigates the methods of solving word problems authored by Diophantus of Alexandria (3rd century CE), the last great mathematician of Antiquity, in his "Arithmetica" treatise.

The article considers the problems with mathematical models, based on systems of nonlinear algebraic equations containing fewer equations than unknowns. The work gives a generalization of Diophantus' approaches to solving some problems from "Arithmetica" treatise. The work derives formulas for an infinite number of solutions, including Diophantine solution.

Generalized solutions are identified using the identity for the sum of the number and the square of half of the difference between denominator and quotient of this number.

The solutions of the system are also presented as linear or quadratic functions with parameter-dependent coefficients. The study obtains the solutions for specific parameter values. Otherwise, the sufficient condition for free terms values is proven, in which all the solutions are integers.

The paper concludes that the methods of solving historical tasks, as well as their generalization, ought to be an important component of training future teachers of mathematics.