Fermat's Last Theorem/Pythagoras

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Introduction[edit]

The origins of numbers is sometimes thought to be lost in the meanders of human history. As a matter of fact since ancient times man has studied numbers and their properties. In the beginning the study was dictated by practical necessity (measures geometrical, astronomical, economical, etc.) but subsequently some humans began to interest themselves in the properties of numbers and sought to understand not only how to solve problems but also why certain formulae or methods always gave the correct result. This desire of abstraction, desire of exploring the more intimate nature of numbers and of their properties saw one of its greatest exponents in Pythagoras.

Pythagoras[edit]

Pythagoras (575 B.C. - 490 B.C.) - mathematician, philosopher, scientist.

During his life Pythagoras passed the years of his youth navigating the length and breadth of the Mediterranean in search of knowledge. During his travels he learnt practically all the ideas in the field of mathematics possessed by the Egyptians and by the Babylonians but, while these peoples were interested principally in the practical applications, Pythagoras wanted to understand the why of mathematics and more generally of things. After some trials and tribulations he succeeded in founding a school of philosophy, this school differently to modern centres of instruction resembled more a sect where numbers were venerated like divine entities. He who entered the school had to divest himself of all his worldly goods which went into the common purse. An obligation of absolute secrecy with respect to the uninitiated was in force and many myths and legends arose around the school.

The theorem of Pythagoras[edit]

Pythagoras is universally famous for his theorem.

Given a and b two sides of a right-angled triangle and c its hypotenuse one has:

a^2 + b^2 = c^2 \,\!

In reality this equation was known by many other mathematicians of the era but Pythagoras became the father of it since he was the first to furnish a generic proof of the equation. He produced by means of a combination of logic and elementary geometry a proof for every right angled triangle. He then passed from an empirical proof for a finite number of cases to a proof as we currently understand it, that is a proof that it is always true for fixed preconditions. Proofs are those which differentiate mathematics from all other sciences. In sciences such as physics, chemistry, etc., the theories are based on theoretical considerations and on experimental trials but they are never considered definitive, they can always be overtaken by the evolution of knowledge. Instead in mathematics once a theorem has been proved, its veracity can no longer be open to discussion. The theorem of Pythagoras was true two thousand years ago and it will be true even in two thousand years from now. The link between Pythagoras’ theorem and Fermat’s last theorem is obvious, it is enough to substitute the power 2 with a generic power n in order to obtain Fermat’s theorem. In fact the theorem of Pythagoras is a particular case of Fermat’s theorem. Fermat was in fact studying the properties of the pythagorean triads (the solutions of the theorem of Pythagoras) when he enunciated his theorem.