Biography

(

)

1620

1629

1631

14

1631

16

1638

1652

1650

1653

I informed you earlier of the death of Fermat. He is alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago.

Fermat, a man of great erudition, has contact with men of learning everywhere. But he is rather preoccupied, he does not report cases well and is confused.

1636

26

1636

I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Viète's analysis could not have sufficed. I will share all of this with you whenever you wish and do so without any ambition, from which I am more exempt and more distant than any man in the world.

Ⓣ ( A course of mathematics )

groping about in the shadows.

Ⓣ ( Discourse on Method )

... seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all.

(

)

1643

1654

1648

1651

x n + y n = z n x^{n} + y^{n} = z^{n} x n + y n = z n

x , y x, y x , y

z z z

n > 2 n > 2 n > 2

I have discovered a truly remarkable proof which this margin is too small to contain.

1670

1993

1993

1994

300

1654

I am delighted to have had opinions conforming to those of M Pascal, for I have infinite esteem for his genius... the two of you may undertake that publication, of which I consent to your being the masters, you may clarify or supplement whatever seems too concise and relieve me of a burden that my duties prevent me from taking on.

Two mathematical problems posed as insoluble to French, English, Dutch and all mathematicians of Europe by Monsieur de Fermat, Councillor of the King in the Parliament of Toulouse.

N x 2 + 1 = y 2 Nx^{2} + 1 = y^{2} N x 2 + 1 = y 2

N N N

(

)

x 2 + 4 = y 3 x^{2} + 4 = y^{3} x 2 + 4 = y 3

x 2 + 2 = y 3 x^{2} + 2 = y^{3} x 2 + 2 = y 3

20

1656

1659

4 k + 1 4k + 1 4 k + 1

4 k + 1 4k + 1 4 k + 1

4 k + 1 4k + 1 4 k + 1

Secretive and taciturn, he did not like to talk about himself and was loath to reveal too much about his thinking. ... His thought, however original or novel, operated within a range of possibilities limited by that [1600 - 1650] time and that [ France ] place.

Recognition of the significance of Fermat's work in analysis was tardy, in part because he adhered to the system of mathematical symbols devised by François Viète, notations that Descartes' "Géométrie" had rendered largely obsolete. The handicap imposed by the awkward notations operated less severely in Fermat's favourite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm.

's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. There is some disputeabout the date of Pierre's birth as given above, since it is possible that he had an elder brotherwho had also been given the name Pierrebut who died young. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery.He attended the University of Toulouse before moving to Bordeaux in the second half of thes. In Bordeaux he began his first serious mathematical researches and inhe gave a copy of his restoration of Apollonius 'sto 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.From Bordeaux Fermat went to Orléans where he studied law at the University. He received a degree in civil law and he purchased the offices of councillor at the parliament in Toulouse. So byFermat was a lawyer and government official in Toulouse and because of the office he now held he became entitled to change his name from Pierre Fermat toFor the remainder of his life he lived in Toulouse but as well as working there he also worked in his home town of Beaumont-de-Lomagne and a nearby town of Castres. From his appointment onMayFermat worked in the lower chamber of the parliament but onJanuaryhe was appointed to a higher chamber, then inhe was promoted to the highest level at the criminal court. Still further promotions seem to indicate a fairly meteoric rise through the profession but promotion was done mostly on seniority and the plague struck the region in the earlys meaning that many of the older men died. Fermat himself was struck down by the plague and inhis death was wrongly reported, then corrected:-The following report, made to Colbert the leading figure in France at the time, has a ring of truth:-Of course Fermat was preoccupied with mathematics. He kept his mathematical friendship with Beaugrand after he moved to Toulouse but there he gained a new mathematical friend in Carcavi . Fermat met Carcavi in a professional capacity since both were councillors in Toulouse but they both shared a love of mathematics and Fermat told Carcavi about his mathematical discoveries.In Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne 's interest was aroused by Carcavi 's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied onApriland, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall, he also told Mersenne about his work on spirals and his restoration of Apollonius 's. His work on spirals had been motivated by considering the path of free falling bodies and he had used methods generalised from Archimedes ' workto compute areas under the spirals. In addition Fermat wrote:-It is somewhat ironical that this initial contact with Fermat and the scientific community came through his study of free fall since Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. This first letter did however contain two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent, his restored text of Apollonius 'sand his algebraic approach to geometryto the Paris mathematicians.His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published, for example Hérigone added a supplement containing Fermat's methods of maxima and minima to his major work. The widening correspondence between Fermat and other mathematicians did not find universal praise. Frenicle de Bessy became annoyed at Fermat's problems which to him were impossible. He wrote angrily to Fermat but although Fermat gave more details in his reply, Frenicle de Bessy felt that Fermat was almost teasing him.However Fermat soon became engaged in a controversy with a more major mathematician than Frenicle de Bessy . Having been sent a copy of Descartes by Beaugrand , Fermat paid it little attention since he was in the middle of a correspondence with Roberval and Étienne Pascal over methods of integration and using them to find centres of gravity. Mersenne asked him to give an opinion onwhich Fermat did, describing it asHe claimed that Descartes had not correctly deduced his law of refraction since it was inherent in his assumptions. To say that Descartes was not pleased is an understatement. Descartes soon found reason to feel even more angry since he viewed Fermat's work on maxima, minima and tangents as reducing the importance of his own workwhich Descartes was most proud of and which he sought to show that hisalone could give. Descartes attacked Fermat's method of maxima, minima and tangents. Roberval and Étienne Pascal became involved in the argument and eventually so did Desargues who Descartes asked to act as a referee. Fermat proved correct and eventually Descartes admitted this writing:-Did this end the matter and increase Fermat's standing? Not at all since Descartes tried to damage Fermat's reputation. For example, although he wrote to Fermat praising his work on determining the tangent to a cycloidwhich is indeed correct Descartes wrote to Mersenne claiming that it was incorrect and saying that Fermat was inadequate as a mathematician and a thinker. Descartes was important and respected and thus was able to severely damage Fermat's reputation.The period fromtowas one when Fermat was out of touch with his scientific colleagues in Paris. There are a number of reasons for this. Firstly pressure of work kept him from devoting so much time to mathematics. Secondly the Fronde, a civil war in France, took place and fromToulouse was greatly affected. Finally there was the plague ofwhich must have had great consequences both on life in Toulouse and of course its near fatal consequences on Fermat himself. However it was during this time that Fermat worked on number theory Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem . This theorem states thathas no non-zero integer solutions forandwhen. Fermat wrote, in the margin of Bachet 's translation of Diophantus 'sThese marginal notes only became known after Fermat's son Samuel published an edition of Bachet 's translation of Diophantus 'swith his father's notes inIt is now believed that Fermat's 'proof' was wrong although it is impossible to be completely certain. The truth of Fermat's assertion was proved in Juneby the British mathematician Andrew Wiles , but Wiles withdrew the claim to have a proof when problems emerged later in. In November Wiles again claimed to have a correct proof which has now been accepted.Unsuccessful attempts to prove the theorem over ayear period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.Fermat's correspondence with the Paris mathematicians restarted inwhen Blaise Pascal Étienne Pascal 's son, wrote to him to ask for confirmation about his ideas on probability Blaise Pascal knew of Fermat through his father, who had died three years before, and was well aware of Fermat's outstanding mathematical abilities. Their short correspondence set up the theory of probability and from this they are now regarded as joint founders of the subject. Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematicians, tried to change the topic from probability to number theory. Pascal was not interested but Fermat, not realising this, wrote to Carcavi saying:-However Pascal was certainly not going to edit Fermat's work and after this flash of desire to have his work published Fermat again gave up the idea. He went further than ever with his challenge problems however:-His problems did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic. The second of the two problems, namely to find all solutions offornot a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution. Brouncker produced rational solutions which led to arguments. Frenicle de Bessy was perhaps the only mathematician at that time who was really interested in number theory but he did not have sufficient mathematical talents to allow him to make a significant contribution.Fermat posed further problems, namely that the sum of two cubes cannot be a cubea special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect, that there are exactly two integer solutions ofand that the equationhas only one integer solution. He posed problems directly to the English. Everyone failed to see that Fermat had been hoping his specific problems would lead them to discover, as he had done, deeper theoretical results.Around this time one of Descartes ' students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat - Descartes correspondence. This led Fermat to look again at the arguments he had usedyears before and he looked again at his objections to Descartes ' optics. In particular he had been unhappy with Descartes ' description of refraction of light and he now settled on a principle which did in fact yield the sine law of refraction that Snell and Descartes had proposed. However Fermat had now deduced it from a fundamental property that he proposed, namely that light always follows the shortest possible path. Fermat's principle, now one of the most basic properties of optics, did not find favour with mathematicians at the time.InFermat had started a correspondence with Huygens . This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and insent to Huygens via Carcavi in, he revealed more of his methods than he had done to others.Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the formcould be written as the sum of two squares. For suppose some number of the formcould not be written as the sum of two squares. Then there is a smaller number of the formwhich cannot be written as the sum of two squares. Continuing the argument will lead to a contradiction. What Fermat failed to explain in this letter is how the smaller number is constructed from the larger. One assumes that Fermat did know how to make this step but again his failure to disclose the method made mathematicians lose interest. It was not until Euler took up these problems that the missing steps were filled in.Fermat is described inasCarl B Boyer, writing in, says:-