Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is credited with early developments that led to infinitesimal calculus, including his adequality. He is also recognized for the discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the then unknown differential calculus, and his research into number theory.
Fermat also made notable contributions to analytic geometry, probability, and optics, andis best known for Fermat’s Last Theorem, which he described in a note at the margin of a copy of Diophantus’ Arithmetica.
Fermat’s pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes’ famous La géométrie. This manuscript was published posthumously in 1679 in “Varia opera mathematica”, as Ad Locos Planos et Solidos Isagoge, (“Introduction to Plane and Solid Loci”).
In his books “Methodus ad disquirendam maximam et minima” and”De tangentibus linearum curvarum”, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature.
Fermat was also the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.
In number theory, Fermat studied Pell’s equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem. He invented a factorization method – Fermat’s factorization method – as well as the proof technique of infinite descent, which he used to prove Fermat’s Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.
Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father’s copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994, using techniques unavailable to Fermat.
Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.
Through his correspondence with Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory. Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in him losing. Fermat subsequently proved why this was the case mathematically.
Fermat’s principle of least time (which he used to derive Snell’s law in 1657) was the first variational principle enunciated in physics since Hero of Alexandria described a principle of least distance in the first century CE. 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 states that no three positive integers a, b, and c can satisfy the equation An + Bn = Cn
If any integer value of n is greater than two. This theorem was first conjectured in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.
No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th Century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof, it was in the Guinness Book of World Records for “most difficult maths problem”.