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Pierre de Fermat | |
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Born | Between 31 October - 6 December 1607^{ [1] } |

Died | (aged 57) | 12 January 1665

Education | University of Orléans (LL.B., 1626) |

Known for | Contributions to number theory, analytic geometry, probability theory Folium of Descartes Fermat's principle Fermat's little theorem Fermat's Last Theorem Adequality Fermat's "difference quotient" method ^{ [2] }(See full list) |

Scientific career | |

Fields | Mathematics and law |

Influences | François Viète, Gerolamo Cardano, Diophantus |

**Pierre de Fermat** (French: [pjɛːʁ də fɛʁma] ) (between 31 October and 6 December 1607^{ [1] } – 12 January 1665) was a French lawyer^{ [3] } at the * Parlement * of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' * Arithmetica *.

The **French** are an ethnic group and nation who are identified with the country of France. This connection may be ethnic, legal, historical, or cultural.

A **parlement**, in the *Ancien Régime* of France, was a provincial appellate court. In 1789, France had 13 parlements, the most important of which was the Parlement of Paris. While the English word *parliament* derives from this French term, parlements were not legislative bodies. They consisted of a dozen or more appellate judges, or about 1,100 judges nationwide. They were the court of final appeal of the judicial system, and typically wielded much power over a wide range of subject matter, particularly taxation. Laws and edicts issued by the Crown were not official in their respective jurisdictions until the parlements gave their assent by publishing them. The members were aristocrats called nobles of the gown who had bought or inherited their offices, and were independent of the King.

**Toulouse** is the capital of the French department of Haute-Garonne and of the region of Occitanie. The city is on the banks of the River Garonne, 150 kilometres from the Mediterranean Sea, 230 km (143 mi) from the Atlantic Ocean and 680 km (420 mi) from Paris. It is the fourth-largest city in France, with 466,297 inhabitants as of January 2014. In France, Toulouse is called the "Pink City".

Fermat was born in the first decade of the 17th century in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a wealthy leather merchant, and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was Claire de Long.^{ [4] } Pierre had one 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 was probably at the Collège de Navarre in Montauban.^{[ citation needed ]}

**Beaumont-de-Lomagne** is a commune in the Tarn-et-Garonne department in the Occitanie region in southern France.

He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius'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 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.

The **University of Orléans** is a French university, in the Academy of Orléans and Tours. As of July 2015 it is a member of the regional university association Leonardo da Vinci consolidated University.

**Bordeaux** is a port city on the Garonne in the Gironde department in Southwestern France.

**Apollonius of Perga** was a Greek geometer and astronomer known for his theories on the topic of conic sections. Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today.

In 1630, he bought the office of a councillor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in six languages (French, Latin, Occitan, 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.

A **Councillor** is a member of a local government council.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends he explored many of the fundamental ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession. Nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus.^{ [5] } Secrecy was common in European mathematical circles at the time. This naturally led to priority disputes with contemporaries such as Descartes and Wallis.^{ [6] }

**Sir Isaac Newton** was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book *Philosophiæ Naturalis Principia Mathematica*, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

**Gottfried Wilhelm** (**von**) **Leibniz** was a prominent German polymath and philosopher in the history of mathematics and the history of philosophy. His most notable accomplishment was conceiving the ideas of differential and integral calculus, independently of Isaac Newton's contemporaneous developments. Mathematical works have generally favored Leibniz's notation as the conventional expression of calculus. It was only in the 20th century that Leibniz's law of continuity and transcendental law of homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of all digital computers.

**René Descartes** was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.

Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."^{ [7] }

**Anders Hald** was a Danish statistician. He was a professor at the University of Copenhagen from 1960 to 1982. While a professor, he did research in industrial quality control and other areas, and also authored textbooks. After retirement, he made important contributions to the history of statistics.

Fermat's pioneering work in analytic geometry (*Methodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum*) was circulated in manuscript form in 1636 (based on results achieved in 1629),^{ [8] } predating the publication of Descartes' famous * La géométrie * (1637), which exploited the work.^{ [9] } This manuscript was published posthumously in 1679 in *Varia opera mathematica*, as *Ad Locos Planos et Solidos Isagoge* (*Introduction to Plane and Solid Loci*).^{ [10] }

In *Methodus ad disquirendam maximam et minimam* and in *De tangentibus linearum curvarum*, Fermat developed a method (adequality) for determining maxima, minima, and tangents to various curves that was equivalent to differential calculus.^{ [11] }^{ [12] } 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 the first person known to have evaluated the integral of general power functions. With his method, he was able to reduce this evaluation to the sum of geometric series.^{ [13] } The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.^{[ citation needed ]}

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 Fermat's 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 right triangle theorem which includes as a corollary 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 proven 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 methods available to Fermat. His famous Last Theorem was first discovered by his son in the margin in his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. It seems that he had not written to Marin Mersenne about it. It was first proven in 1994, by Sir Andrew Wiles, 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.^{[ citation needed ]}

Through their correspondence in 1654, Fermat and Blaise Pascal helped lay the foundation 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.^{ [14] } 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 his losing. Fermat showed mathematically why this was the case.^{ [15] }

The first variational principle in physics was articulated by Euclid in his *Catoptrica*. It says that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path gave the shortest length and the least time.^{ [16] } Fermat refined and generalized this to "light travels between two given points along the path of shortest *time*" now known as the * principle of least time *.^{ [17] } For this, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The terms Fermat's principle and *Fermat functional* were named in recognition of this role.^{ [18] }

Pierre de Fermat died on January 12, 1665 at Castres, in the present-day department of Tarn.^{ [19] } The oldest and most prestigious high school in Toulouse is named after him: the Lycée Pierre-de-Fermat . French sculptor Théophile Barrau made a marble statue named *Hommage à Pierre Fermat* as a tribute to Fermat, now at the Capitole de Toulouse.

Together with René Descartes, 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."^{ [20] }

Regarding Fermat's work in analysis, Isaac Newton wrote that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."^{ [21] }

Of Fermat's number theoretic work, the 20th-century mathematician André Weil wrote 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 descent which is rightly regarded as Fermat's own."^{ [22] } 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 properties of the rational points on a standard cubic."^{ [23] } 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.

- 1 2 When was Pierre de Fermat Born? Mathematical Association of America webpage with references
- ↑ Donald C. Benson,
*A Smoother Pebble: Mathematical Explorations*, Oxford University Press, 2003, p. 176. - ↑ W.E. Burns, The Scientific Revolution: An Encyclopedia, ABC-CLIO, 2001, p. 101
- ↑ "When Was Pierre de Fermat Born? | Mathematical Association of America".
*www.maa.org*. Retrieved 2017-07-09. - ↑ Larson, Hostetler, Edwards (2008).
*Essential Calculus Early Transcendental Functions*. U.S.A: Richard Stratton. p. 159. ISBN 978-0-618-87918-2.CS1 maint: Multiple names: authors list (link) - ↑ Ball, Walter William Rouse (1888).
*A short account of the history of mathematics*. General Books LLC. ISBN 978-1-4432-9487-4. - ↑ Faltings, Gerd (1995), "The proof of Fermat's last theorem by R. Taylor and A. Wiles" (PDF),
*Notices of the American Mathematical Society*,**42**(7): 743–746, MR 1335426 - ↑ Daniel Garber, Michael Ayers (eds.),
*The Cambridge History of Seventeenth-century Philosophy, Volume 2*, Cambridge University Press, 2003, p. 754 n. 56. - ↑ "Pierre de Fermat | Biography & Facts".
*Encyclopedia Britannica*. Retrieved 2017-11-14. - ↑ Gullberg, Jan.
*Mathematics from the birth of numbers*, W. W. Norton & Company; p. 548. ISBN 0-393-04002-X ISBN 978-0393040029 - ↑ Pellegrino, Dana. "Pierre de Fermat" . Retrieved 2008-02-24.
- ↑ Florian Cajori, "Who was the First Inventor of Calculus" The American Mathematical Monthly (1919) Vol.26
- ↑ Paradís, Jaume; Pla, Josep; Viader, Pelegrí (2008), "Fermat's method of quadrature",
*Revue d'Histoire des Mathématiques*,**14**(1): 5–51, MR 2493381^{[ permanent dead link ]} - ↑ O'Connor, J. J.; Robertson, E. F. "The MacTutor History of Mathematics archive: Pierre de Fermat" . Retrieved 2008-02-24.
- ↑ Eves, Howard.
*An Introduction to the History of Mathematics*, Saunders College Publishing, Fort Worth, Texas, 1990. - ↑ Kline, Morris (1972).
*Mathematical Thought from Ancient to Modern Times*. New York: Oxford University Press. pp. 167–168. ISBN 978-0-19-501496-9. - ↑ "Fermat's principle for light rays". Archived from the original on March 3, 2016. Retrieved 2008-02-24.
- ↑ Červený, V. (July 2002). "Fermat's Variational Principle for Anisotropic Inhomogeneous Media".
*Studia Geophysica et Geodaetica*.**46**(3): 567. doi:10.1023/A:1019599204028.^{[ dead link ]} - ↑ Klaus Barner (2001):
*How old did Fermat become?*Internationale Zeitschrift für Geschichte und Ethik der Naturwissenschaften, Technik und Medizin. ISSN 0036-6978. Vol 9, No 4, pp. 209-228. - ↑ Bernstein, Peter L. (1996).
*Against the Gods: The Remarkable Story of Risk*. John Wiley & Sons. pp. 61–62. ISBN 978-0-471-12104-6. - ↑ Simmons, George F. (2007).
*Calculus Gems: Brief Lives and Memorable Mathematics*. Mathematical Association of America. p. 98. ISBN 978-0-88385-561-4. - ↑ Weil 1984, p.104
- ↑ Weil 1984, p.105

- Weil, André (1984).
*Number Theory: An approach through history From Hammurapi to Legendre*. Birkhäuser. ISBN 978-0-8176-3141-3.

- Barner, Klaus. "Pierre de Fermat (1601? - 1665): His life besides mathematics".
*Newsletter of the European Mathematical Society, December 2001, Pp. 12-16*. - Mahoney, Michael Sean (1994).
*The mathematical career of Pierre de Fermat, 1601 - 1665*. Princeton Univ. Press. ISBN 978-0-691-03666-3. - Singh, Simon (2002).
*Fermat's Last Theorem*. Fourth Estate Ltd. ISBN 978-1-84115-791-7.

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- Pierre de Fermat at the
*Encyclopædia Britannica* - Fermat's Achievements
- Fermat's Fallibility at MathPages
- The Correspondence of Pierre de Fermat in EMLO
- History of Fermat's Last Theorem (French)
- The Life and times of Pierre de Fermat (1601 - 1665) from W. W. Rouse Ball's History of Mathematics
- O'Connor, John J.; Robertson, Edmund F., "Pierre de Fermat",
*MacTutor History of Mathematics archive*, University of St Andrews .

In mathematics, a **conjecture** is a conclusion or proposition based on incomplete information, for which no proof or disproof has yet been found. Conjectures such as the Riemann hypothesis or Fermat's Last Theorem have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.

**Diophantus of Alexandria** was an Alexandrian Hellenistic mathematician, who was the author of a series of books called *Arithmetica*, many of which are now lost. Sometimes called "the father of algebra", his texts deal with solving algebraic equations. While reading Claude Gaspard Bachet de Méziriac's edition of Diophantus' *Arithmetica,* Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted in the margin without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as *adaequalitas* in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought.

The area of study known as the **history of mathematics** is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, together with Ancient Egypt and Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy and to formulate calendars and record time.

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Number theory** is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

**Fermat's little theorem** states that if p is a prime number, then for any integer a, the number *a*^{p} − *a* is an integer multiple of p. In the notation of modular arithmetic, this is expressed as

**Gorō Shimura** was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

**Claude Gaspard Bachet de Méziriac** was a French mathematician, linguist, poet and classics scholar born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy.

**Discrete mathematics** is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently discovered calculus in the mid-17th century. However, both inventors claimed that the other had stolen his work, and the Leibniz-Newton calculus controversy continued until the end of their lives.

In additive number theory, the **Fermat polygonal number theorem** states that every positive integer is a sum of at most nn-gonal numbers. That is, every positive integer 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 five or fewer pentagonal numbers, and so on.

**Gerhard Frey** is a German mathematician, known for his work in number theory. His Frey curve, a construction of an elliptic curve from a purported solution to the Fermat equation, was central to Wiles's proof of Fermat's Last Theorem.

In mathematics, **arithmetic geometry** is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

The **Fermat prize** of mathematical research bi-annually rewards research works in fields where the contributions of Pierre de Fermat have been decisive:

In number theory **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2. The cases *n* = 1 and *n* = 2 have been known since antiquity to have an infinite number of solutions.

A timeline of **number theory**.

This is a timeline of pure and applied mathematics history.

**Wiles's proof of Fermat's Last Theorem** is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.

**Adequality** is a technique developed by Pierre de Fermat in his treatise *Methodus ad disquirendam maximam et minimam* to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.". Diophantus coined the word παρισότης (*parisotēs*) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as *adaequalitas*. Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words *adéquation* and *adégaler*.

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