Pierre de Fermat

# Pierre de Fermat

Overview
Pierre de Fermat was a French
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...

lawyer at the Parlement
Parlement
Parlements were regional legislative bodies in Ancien Régime France.The political institutions of the Parlement in Ancien Régime France developed out of the previous council of the king, the Conseil du roi or curia regis, and consequently had ancient and customary rights of consultation and...

of Toulouse
Toulouse
Toulouse is a city in the Haute-Garonne department in southwestern FranceIt lies on the banks of the River Garonne, 590 km away from Paris and half-way between the Atlantic Ocean and the Mediterranean Sea...

, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

In the history of infinitesimal calculus, adequality is a technique developed by Pierre de Fermat. Fermat said he borrowed the term from Diophantus. Adequality was a technique first used to find maxima for functions and then adapted to find tangent lines to curves...

. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...

s of curved lines, which is analogous to that of the then unknown differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, and his research into number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. He made notable contributions to analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, and optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

. He is best known for Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

, which he described in a note at the margin of a copy of Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...

' Arithmetica
Arithmetica
Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.Equations in the book are called Diophantine equations...

.
Discussion
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Quotations

a^p \equiv a \pmod\,\!

Also known as Fermat's little theorem|Fermat's little theorem, the formula above is true for any prime p where a

$a^p \equiv a \pmod\,\!$

Also known as Fermat's little theorem|Fermat's little theorem, the formula above is true for any prime $p$ where $a$

$a^p \equiv a \pmod\,\!$

Also known as Fermat's little theorem|Fermat's little theorem, the formula above is true for any prime $p$ where
Encyclopedia
Pierre de Fermat was a French
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...

lawyer at the Parlement
Parlement
Parlements were regional legislative bodies in Ancien Régime France.The political institutions of the Parlement in Ancien Régime France developed out of the previous council of the king, the Conseil du roi or curia regis, and consequently had ancient and customary rights of consultation and...

of Toulouse
Toulouse
Toulouse is a city in the Haute-Garonne department in southwestern FranceIt lies on the banks of the River Garonne, 590 km away from Paris and half-way between the Atlantic Ocean and the Mediterranean Sea...

, France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

In the history of infinitesimal calculus, adequality is a technique developed by Pierre de Fermat. Fermat said he borrowed the term from Diophantus. Adequality was a technique first used to find maxima for functions and then adapted to find tangent lines to curves...

. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinate
Ordinate
In mathematics, ordinate refers to that element of an ordered pair which is plotted on the vertical axis of a two-dimensional Cartesian coordinate system, as opposed to the abscissa...

s of curved lines, which is analogous to that of the then unknown differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

, and his research into number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

. He made notable contributions to analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, and optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

. He is best known for Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

, which he described in a note at the margin of a copy of Diophantus
Diophantus
Diophantus of Alexandria , sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost...

' Arithmetica
Arithmetica
Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.Equations in the book are called Diophantine equations...

.

## Life and work

Fermat was born in Beaumont-de-Lomagne
Beaumont-de-Lomagne
Beaumont-de-Lomagne is a commune in the Tarn-et-Garonne department in the Midi-Pyrénées region in southern France.-History:Beaumont-de-Lomagne, bastide, was founded in 1276 following the act of coregency between the abbey of Grandselve and King Philip III of France - the King was represented by his...

, Tarn-et-Garonne
Tarn-et-Garonne
Tarn-et-Garonne is a French department in the southwest of France. It is traversed by the Rivers Tarn and Garonne, from which it takes its name.-History:...

, France; the late 15th century mansion where Fermat was born is now a museum. He was of Basque
Basque people
The Basques as an ethnic group, primarily inhabit an area traditionally known as the Basque Country , a region that is located around the western end of the Pyrenees on the coast of the Bay of Biscay and straddles parts of north-central Spain and south-western France.The Basques are known in the...

origin. Fermat's father was a wealthy leather merchant and second consul of Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been at the local Franciscan
Franciscan
Most Franciscans are members of Roman Catholic religious orders founded by Saint Francis of Assisi. Besides Roman Catholic communities, there are also Old Catholic, Anglican, Lutheran, ecumenical and Non-denominational Franciscan communities....

monastery.
He attended the University of Toulouse
University of Toulouse
The Université de Toulouse is a consortium of French universities, grandes écoles and other institutions of higher education and research, named after one of the earliest universities established in Europe in 1229, and including the successor universities to that earlier university...

before moving to Bordeaux
Bordeaux
Bordeaux is a port city on the Garonne River in the Gironde department in southwestern France.The Bordeaux-Arcachon-Libourne metropolitan area, has a population of 1,010,000 and constitutes the sixth-largest urban area in France. It is the capital of the Aquitaine region, as well as the prefecture...

in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius
Apollonius of Perga
Apollonius of Perga [Pergaeus] was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes...

's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...

.

From Bordeaux, Fermat went to Orléans
Orléans
-Prehistory and Roman:Cenabum was a Gallic stronghold, one of the principal towns of the Carnutes tribe where the Druids held their annual assembly. It was conquered and destroyed by Julius Caesar in 52 BC, then rebuilt under the Roman Empire...

where he studied law at the University. He received a degree in civil law before, in 1631, receiving the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life. Due to the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque, classical Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

and Wallis. He developed a close relationship with Blaise Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

.

Anders Hald
Anders Hald
Anders Hald was a Danish statistician who made contributions to the history of statistics.He was a professor at the University of Copenhagen from 1960 to 1982.- Bibliography :...

writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebra
New algebra
The new algebra or symbolic analysis is a formalization of algebra promoted by François Viète in 1591 and by his successors...

ic methods."

### Work

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 Methodus ad disquirendam maximam et minima and in 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
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...

.

Fermat was 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
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

, and then Leibniz, when they independently developed the fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...

.

In number theory, Fermat studied Pell's equation
Pell's equation
Pell's equation is any Diophantine equation of the formx^2-ny^2=1\,where n is a nonsquare integer. The word Diophantine means that integer values of x and y are sought. Trivially, x = 1 and y = 0 always solve this equation...

, perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

s, amicable number
Amicable number
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. A pair of amicable numbers constitutes an aliquot sequence of period 2...

s and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...

. He invented a factorization method—Fermat's factorization method
Fermat's factorization method
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares:N = a^2 - b^2.\...

—as well as the proof technique of infinite descent
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...

, 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
Fermat polygonal number theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive number can be written as the sum of three or fewer triangular numbers, and as the sum of four or fewer square numbers, and as the sum of...

, which states that each number is a sum of three triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

s, four square numbers
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...

, five pentagonal number
Pentagonal number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical...

s, and so on.

Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

, 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
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....

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
Marin Mersenne
Marin Mersenne, Marin Mersennus or le Père Mersenne was a French theologian, philosopher, mathematician and music theorist, often referred to as the "father of acoustics"...

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 equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

s, and he looked for all possible general solutions. He often proved that certain equations had no solution
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

, 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
Problem of points
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is...

, they are now regarded as joint founders of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

. 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
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...

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
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...

in 1657) was the first variational principle
History of variational principles in physics
A variational principle in physics is an alternative method for determining the state or dynamics of a physical system, by identifying it as an extremum of a function or functional...

enunciated in physics since Hero of Alexandria
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...

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
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...

in physics. The terms Fermat's principle
Fermat's principle
In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light...

and Fermat functional were named in recognition of this role.

### Death

He died at Castres
Castres
Castres is a commune, and arrondissement capital in the Tarn department and Midi-Pyrénées region in southern France. It lies in the former French province of Languedoc....

, Tarn. The oldest and most prestigious high school in Toulouse
Toulouse
Toulouse is a city in the Haute-Garonne department in southwestern FranceIt lies on the banks of the River Garonne, 590 km away from Paris and half-way between the Atlantic Ocean and the Mediterranean Sea...

is named after him: the Lycée Pierre de Fermat. French sculptor Théophile Barrau
Théophile Barrau
Théophile Barrau was a French sculptor.Barrau was born in Carcassonne. He was a student of Alexandre Falguière and started at the Salon in 1874. He received awards in 1879, 1880, 1889, and became a Chevalier of the Legion of Honor in 1892...

made a marble statue named Hommage à Pierre Fermat as tribute to Fermat, now at the Capitole of Toulouse.

## Assessment of his work

Together with René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

, Fermat was one of the two leading mathematicians of the first half of the 17th century. According to Peter L. Bernstein, in his book Against the Gods, Fermat "was a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. In the course of what turned out to be an extended correspondence with Pascal, he made a significant contribution to the theory of probability. But Fermat's crowning achievement was in the theory of numbers."

Regarding Fermat's work in analysis, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."

Of Fermat's number theoretic work, the great 20th-century mathematician André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

wrote that "... what we possess of his methods for dealing with curves
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

of genus 1
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

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 descent
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...

which is rightly regarded as Fermat's own." Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

properties of the rational point
Rational point
In number theory, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equationsthen the K-rational points are solutions ∈Kn of the equations...

s on a standard cubic." 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.