The analytic geometry. A contemporary of René Descartes, he independently came up with a three-dimensional geometry, but did not publish his work, and the field became known as Cartesian geometry.
The number theory. His brilliant research entitled him to rank as the founder of the modern number theory.
He stated the Fermat's Last Theorem (1637) as well as “Fermat's little theorem”( 1640), and 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. 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. In 1654, Blaise Pascal wrote a letter asking about Fermat's views on probability. Their series of correspondences became the foundation of probability theory.
In 1660 he planned to meet with Pascal, but meeting not held due to ill of both scientists.
Optics. He 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.
In the second half of the 1620s. he moved to Bordeaux where he began his first serious mathematical researches. From Bordeaux Fermat went to Orléans where he studied law at the University. He became a councillor at the parliament in 1631 and receved the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life.
In 1648 he was promoted to king's counselor in the Parliament of Toulouse. He quickly moved up the ranks and in 1652, he became the chief magistrate of the criminal court. His work allowed to spend large amounts of time in isolation.
It was during this time, he obeying his passion for mathematics developed the fundamental theorem and theories.
In 1631 he married his mother's cousin, Louise de Long; they had three sons and two daughters. Contemporaries described him as an honest, accurate, balanced and genial man, a brilliantly erudite in mathematics and in the humanities, a connoisseur of many ancient and living languages, in which he wrote good poetry.
He died on January 12, 1665 in Castres, France.