Mathematical Work and Discoveries

Despite Fermat's obvious aptitude for the field, he treated math more like a hobby, describing his discoveries to friends in letters rather than writing his proofs in papers for publication. He only published one mathematical work in his lifetime, “Concerning the Comparison of Curved Lines with Straight Lines," as a supplement to the Veterum Geometria Promota by the mathematician Antoine de La Loubère.

 

Despite this, Fermat's considerable body of mathematical work was circulated and has endured. His accomplishments include:

 

Number Theory: Fermat's research into Number Theory has led some to call him the founder of modern number theory. "Fermat's Little Theorem" (1640) and "Fermat's Last Theorem" (1637) are related to  this.  In his number theory work, he also developed the inductive “infinite descent method," which was  the first general proof of diophantine questions. He made some discoveries in regard to the properties of numbers, on which he afterwards built his method of calculating probabilities. 

Mathematical analysis: He discovered  an original method for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus.

 

Physics: He obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.

The theory of probability: Through correspondence with Blaise Pascal, a fellow French mathematician, the two laid the groundwork for probability theory,  the branch of mathematics concerned with probability, the analysis of random phenomena.

Optics: Fermat discovered the least time principle which states that light will travel through an optical system in such a way as to pass from starting to ending point in the least amount of time  Fermat's principle of least time  was the first variational principle enunciated in physics. 
In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The term "Fermat functional" was named in recognition of this role. 

 

Fermat's Last Theorem

Sometime around 1637, Fermat made the following note in a copy of Arithmetica by the Greek mathematician Diophantus:

 

                                            It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general,

                                            any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this,

                                            which this margin is too narrow to  contain.

 

Following his death in 1665, his son, Clement-Samuel Fermat, reprinted Arithmetica with his father's notes. The mysterious hint about his "marvellous proof" would go on to captivate minds of mathematicians everywhere, who struggled for years to replicate the proof.

 

It's unknown whether Fermat actually had the valid proof he described, although popular scholastic consensus is that it is unlikely. This is partially based on the fact that the mathematic knowledge available to Fermat at the time was not sophiscicated enough, as well as the fact that Fermat never mentions the proof again. It's also possible that he did find what he thought was a valid proof, only to discover errors in it, but again, this is pure conjecture.

 

Throughout the 19th and 20th centuries, a number of mathematicians tackled Fermat's Last Theorem, including Sophie Germain, Louis Mordell, Ernst Kummer, Harry Vandiver, Samuel Wagstaff and Gerhard Frey. However, it wasn't until the mid-1990s that British mathematician Andrew Wiles, working off conjectures by other mathematicians, discovered a successful proof.

 

The story of Wiles' discovery is described in the video below: