Solving Selected Linear Diophantine Equations and Fermat's Last Theorem
Solving Selected Linear Diophantine Equations and Fermat's Last Theorem

Author(s): Tomáš Lengyelfalusy, Viliam Ďuriš
Subject(s): Philosophy of Science
Published by: Masarykova univerzita nakladatelství
Keywords: linear diophantine equations; number theory; Matlab; Fermat's Last Theorem;
Summary/Abstract: Instead of his age (84), the epitaph on Diophantine's tomb states a polynomial expression with one unknown, which represents his age: "This is Diophantine's tomb. The inscription on the tomb reveals his age. God vouchsafed that he should be a boy for the sixth part of his life. When a twelfth was added, his cheeks acquired a beard. He kindled for him the light of marriage after a seventh. In the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life." Diophantus was one of the first mathematicians who significantly contributed to the number theory, and particularly to the solution of equations. This paper deals with a special type of equations – the so-called linear diophantine equations with two unknowns – which we can use to solve many practical problems in various areas of contemporary mathematics. In the current paper, we present some interesting examples of these issues and their possible implementation into the Matlab computing environment, which can be used when teaching the number theory and discrete mathematics. The paper also discusses probably the most important diophantine equation in the history of mathematics – the Fermat's Last Theorem – which was introduced by the French lawyer and mathematician Pierre de Fermat, and verified only at the end of the 20th century.

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