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Pierre de Fermat

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.
Buste 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 fellow 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]


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.


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]

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.

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.

Pakistan Independence Day

Pakistan's independence day (also known as Yom-e-Azaadi) is observed on 14 August, the day before Pakistan was made an independent country based on border lines created by the British during the end of their rule of India. Pakistan became an independent country in 1947. The day is a national holiday in Pakistan. The day is celebrated all over the country with flag raising ceremonies, tributes to the national heroes and fireworks taking place in the capital, Islamabad. The main celebrations takes place in Islamabad, where the President and Prime Minister raise the national flag at the Presidential and Parliament buildings and deliver speeches that are televised live. In the speech, the leaders highlight the achievements of the government, goals set for the future and in the words of the father of the nation, Quaid-e-Azam, bring "Unity, Faith and Discipline" to its people.
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{{Main|History of Pakistan|Pakistan Movement|Partition of India}5 Independence of Pakistan was officially announced on Thursday, August 15, 1947 at 12:00 a.m (midnight). but the ceremony itself took place half an hour early on the actual day.[citation needed] This day is celebrated 1 day earlier in comparison to India even though independence was granted by the British Raj to both countries at midnight on 15 August 1947.

Independence Day was not celebrated on 14 August 2010 as a show of moral support to those affected by the 2010 Pakistan floods. All official Military and Government events that are traditionally held on this day were also scrapped to divert the expenditure incurred by these events to flood relief.[1]
[edit] Celebrations
The Minar-e-Pakistan fully lit to commemorate the independence of Pakistan from the British Empire
The change of guard ceremony takes place at various monuments throughout the country. Here the Pakistan Navy cadets salute the tomb of the father of the nation, Mohammed Ali Jinnah
Students lighting candles at midnight to bring luck and hope for the future.

14 August is a National holiday of Pakistan. In the capital Islamabad and in all major cities of Pakistan the Government Offices are lit up as well as all the larger skyscrapers. Flag hoisting ceremonies and cultural programs take place in all the provincial capitals. In the cities around the country the Flag Hoisting Ceremony is done by the Nazim (Mayor) belonging to that constituency. In various private organisations the Flag Hoisting Ceremony is carried out by a Senior officer of that organisation. Schools and colleges around the country organise flag hosting ceremony and various cultural activities within their respective premises. Families and friends get together for lunch or dinner, or for an outing. Housing colonies, cultural centres, and societies hold entertainment programmes and competitions.

Other events include: Changing of the guard at the mausoleum of Mohammed Ali Jinnah, Mazar-e-Quaid, Wagah Border ceremonies, fashion and musical concerts, both sides releasing prisoners that may have crossed each others borders.