in 1637, Pierre de Fermat, a judge at the French Parliament of Toulouse, wrote in the margin of a book on Diophantine equations that he had devised a marvelous proof that there are no positive whole number solutions to the equation xn + yn = zn for n greater than two. His proof, for n = 4 is known, but his general proof for all n greater than two was never found.
It is an interesting aside that Fermat was not a professional mathematician. He did not publish his findings, he simply conveyed them in letters to other mathematicians, and thus was considered an amateur. The less than modest French mathematician and philosopher Rene Descartes tried to discredit Fermat by proclaiming that he was “a troublemaker who owed his reputation to a few lucky guesses”. However, in one dispute after another, e.g. their derivations of the sine law for the refraction of light, Fermat proved to be right and Descartes wrong. While Descartes clearly considered himself to be the superior intellect of the day, a comparison of their works reveals the fact that Fermat was the better scientist and mathematician of the two.
Descartes’ arrogance shines out in the following statement: “I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.” Implying that he could have explained much more. This is analogous to a classmate of mine who liked to say “I’m not conceited, I’m actually twice as smart as I say I am!”
Regarding Fermat's work, Sir Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."
Speaking of Fermat's work in number theory, Mathematician Andre Weil says that: "what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the method of descent which is rightly regarded as Fermat's own. … With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.
Never-the-less, popular history has treated Rene Descartes very well, while ignoring Fermat. The name Descartes is well known to every student of mathematics and science, while Fermat’s name is virtually unknown except for in relation to Fermat’s Last Theorem, -and most modern mathematicians openly doubt that he actually proved it! Why? Because for more than 300 years, professional mathematicians tried to find a proof, and failed.
The power of Fermat’s math lies in his method of infinite descent, and the principle of “efficient purpose in nature”, and in 1964-5, while teaching high school mathematics, using the same simple principles used by Pierre de Fermat, I produced a proof which I first submitted to a professional mathematician in 1966. Because my proof was completed in 1965, I call it FLT65.