A mathematical problem awaited a solution for over 300 years.

Around 1637, the Frenchman Pierre de Fermat wrote in the margin of the book "Arithmetica" by the mathematician Diophantus (who lived in the Mondays century): "The equation A ^N + B ^N = C ^N has no integer solutions if N is an integer greater than 2. I have found a truly marvelous proof of this fact, but the margin is too narrow to contain it."

This note was found after his death, and for more than three centuries, mathematicians bitterly lamented that the margin wasn't larger, as no one was able to find the proof that Fermat claimed to have.

The root of the problem lies in the Pythagorean theorem, which we all know from school: "in any right triangle, if A and B are the lengths of the shorter sides –the legs– and C is the length of the longest side –the hypotenuse– then ^A² + ^B² = ^C² ".

This theorem is named after the Greek philosopher Pythagoras (c. 570-495 BC) because he is believed to have been the first to prove it mathematically. But the statement was known long before, by the great civilizations of Mesopotamia and the Indus Valley.

It is interesting to study the equation ^A² + ^B² = ^C² in the special case where A, B, and C are integers. We currently know that there is an infinite set of such solutions, for example, (A = 3, B = 4, C = 5) and (A = 5, B = 12, C = 13).

"Arithmetica" is a collection of thirteen texts by Diophantus on equations of this type, where the solutions are integers. Fermat was not the only one to make annotations in its margins. The Byzantine scholar Janus Chortasmenos (1370-1437) wrote on the same page: "May your soul be with Satan, Diophantus, for the difficulty of your theorems, especially this one here."

Although most of these texts have been lost, a portion was translated into Latin and published in Europe from the late 1Fridays century onwards, becoming very influential.

Pierre de Fermat was born in the early 17th century and died in 1665. He was a judge and mathematician, and displayed extraordinary erudition in many other subjects. Among his most important discoveries is the law – Fermat's principle – according to which light travels from one point to another along the path that minimizes the duration of the journey. This is at the origin of one of the most fundamental laws of physics: the principle of least action (or "law of least effort," in colloquial language).

Fermat also made important discoveries in calculus – Newton acknowledged being inspired by him – and is considered the creator of modern number theory. He himself proved the case N = 4 of his famous statement, and other particular cases were proven later. But the attempt to prove the general case, that is, to find an argument that holds for all values of N greater than 2, resisted the efforts of many of the best mathematicians for over 300 years.

In 1983, the German physicist Gerd Faltings (born in 1954) proved that if there are integer solutions to the equation ^Aₙ₋₁ + ^Bₙ₋₁ = ^Cₙ₋₁ , then essentially they are finite in number. For this work, Faltings won the Fields Medal, the most prestigious award in mathematics, in 1986.

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