****** ***** ***
Pierre de Fermat
*** ***** ***** ***
Born August 17, 1601
Beaumont-de-Lomagne, France
Died January 12, 1665 (aged 63)
Castres, France
Residence France
Nationality French
Fields Mathematics and Law
Known for Number theory
Analytic geometry
Fermat's principle
Probability
Fermat's Last Theorem
Influences François Viète
* *** **** ******* *
Pierre de Fermat
* ***** ********* *
French pronunciation: [pjɛːʁ dəfɛʁˈma];
17[1] August 1601 or 1607/8[2] – 12 January 1665) was a French
lawyer at the Parlement of Toulouse, France, and an amateur
mathematician who is given credit for early developments that led
to infinitesimal calculus, including his 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 the then unknown
differential calculus, and his research into number theory. He
made notable contributions to analytic geometry, probability, and
optics. He is best known for Fermat's Last Theorem, which he
described in a note at the margin of a copy of Diophantus'
Arithmetica.
Contents
1 Life and work
1.1 Work
1.2 Death
2 Assessment of his work
3 See also
4 Notes
4.1 Books referenced
5 Further reading
6 External links
Life and work
Fermat was born in Beaumont-de-Lomagne, Tarn-et-Garonne, France;
the late 15th century mansion where Fermat was born is now a
museum. He was of Basque 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
monastery.
Bust in the Salle des Illustres in Capitole de Toulouse
He attended the University of Toulouse before moving to Bordeaux
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'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.
From Bordeaux, Fermat went to Orléans 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[citation needed], 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 and Wallis. He developed a close
relationship with Blaise Pascal.[3]
Anders Hald writes that, "The basis of Fermat's mathematics was
the classical Greek treatises combined with Vieta's new algebraic
methods."[4]
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").[5]
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.[6] 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.
Pierre de Fermat
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.[7] 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 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.[8] 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.[9]
Fermat's principle of least time (which he used to derive Snell's
law in 1657) was the first variational principle[10] 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.[11]
Death
Plaque at the place of burial of Pierre de Fermat
Place of burial of Pierre de Fermat in Place Jean Jaurés,
Castres, France. Translation of the plaque: in this place was
buried on January 13, 1665, Pierre de Fermat, councilor of the
chamber of Edit and mathematician of great renown, celebrated for
his theorem (sic),
an + bn ≠ cn for n>2
He died at Castres, Tarn.[2] 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 tribute to Fermat, now at the
Capitole of Toulouse.
Assessment of his work
Holographic will handwritten by Fermat on 4 March 1660 — kept
at the Departmental Archives of Haute-Garonne, in 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."[12]
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."[13]
Of Fermat's number theoretic work, the great 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."[14]
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."[15] 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.
See also
Diagonal form
Euler's theorem
Fermat cubic
Fermat Prize
Fermat pseudoprime
Fermat quotient
Fermat's spiral
Fermat's theorem (stationary points)
dogmeat