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Diophantine Equations I


From ancient times it has been known that the search for all positive integers that satisfy the identity x² + y² = z² is equivalent to the problem of determining all right triangles whose sides are integers. Such suits are called Pythagorean and the equation x² + y² = z², whose solutions are positive integers, is called the Diophantine equation. This equation can be generalized to xⁿ + yⁿ = zⁿ, where no is a natural number and no > 2. It is known as Fermat's equation, the most famous in mathematics and will be the subject of some of our reflections. Another famous example of the Diophantine equation is the Fibonacci curve, that is, the system of equations x² + y² = z², x² - y² = t², which was first studied in 1220 by Leonardo de Pisa better known as Fibonacci.

The Greek mathematician Diophantus of Alexandria (4O Century BC) was the first to investigate the problem of determining whole solutions of equations, particularly cases where the number of variables is greater than the number of equations. Diophantus was content to find a single solution instead of all solutions, and allowed fractional solutions instead of whole solutions. However, this distinction is irrelevant: for example, consider the equation x² + y² = z². If a fractional solution is obtained, then from it a whole solution is obtained. Conversely, if an entire solution is found, one gets a fractional from it. For example, from solution (3,4,5) we get solution (3 / 6,4 / 6,5 / 6), that is, (1 / 2,2 / 3,5 / 6) and vice versa finding the least common multiple. It is due to Diophantus the fundamental idea of ​​studying whole solutions, and some basic theorems, about the representation of numbers as the sum of squares, of which he partially knew the statements, and others whose statements he did not know.

The study of Diophantine equations is one of the most beautiful and interesting, and also one of the most difficult, for in its essence lie the deep and subtle links that Number Theory maintains with Logic, Algebraic Geometry, and Theory of Diophantine Approaches. On the other hand, there is no general method that decides whether or not an arbitrary equation has integer solutions, or a method that establishes how many solutions the equation allows. In 1900, mathematician David Hilbert (1862-1943), one of the most prestigious and influential of his day, gave a speech at the International Congress of Mathematics in Paris, where he announced the problems that were, in his view, the most important in the century. XX. Among these problems, the tenth concerned the investigation of an algorithm that would allow, in a finite number of steps, to decide whether an arbitrary Diophantine equation has a solution and, if so, what is the number of solutions. In the following years, several mathematicians intensely investigated the existence of such an unsuccessful algorithm and then began to doubt the existence of such an algorithm. In 1961 Martin Davis, Hillary Putnam, and Julia Robinson, using Logic and Number Theory, demonstrated that such an algorithm could not exist, but assumed that a certain hypothesis was valid. In 1970, the young Russian mathematician Y. Matijasievič demonstrated that such a hypothesis was true. Thus, it was demonstrated that such an algorithm could not in fact exist. What is interesting here is that such an algorithm, which serves to solve all Diophantine equations, does not exist, that is, an algorithm that decides for all equations, in a finite number of steps, whether or not it has a solution. But that does not prevent us from eventually finding solutions, or even all solutions of a certain equation, such as the equation x² + y² = z². Therefore, there is no common recipe for solving all Diophantine equations. Each equation has its own specificity, which partly explains why this area of ​​research is so difficult.

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