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Pierre de Fermat (piyer dö ferma okunur) (d. 1601, Beaumont-de-Lomagne – ö. 12 Ocak 1665, Castres), Bask kökenli Fransız hukukçu ve matematikçi. İlk öğrenimini doğduğu şehirde yapmıştır. Yargıç olmak için çalışmalarına Toulouse’de devam etmiştir. Fermat, memurluğunun yoğun işlerinden geriye kalan zamanlarında matematikle uğraşmıştır. Arşimet'in eğildiği diferansiyel hesaba geometrik görünümle yaklaşmıştır. Sayılar teorisinde önemli sonuçlar bulmuş, olasılık ve analitik geometriye de katkılarda bulunmuştur.
Günümüzde hatırlanmasının en önemli sebebi Fermat'nın Son Teoremi'dir. Modern sayılar kuramının kurucusu olarak kabul edilen 17. yüzyıl matematikçisi Pierre de Fermat'nın adını taşıyan bu teorem, şu şekilde ifade edilebilir:
Herhangi x, y, ve z pozitif tam sayıları için,
ifadesini sağlayan ve 2'den büyük bir doğal sayı n yoktur. Fermat, bu problemi çözmüş, kanıtı da Eski Yunanlı matematikçi Diaphontos'un Arithmetika adlı kitabının kendindeki kopyasının sayfalarından birinin kenarına 1637'de şöyle yazmıştı:
x, y, z ve n pozitif tamsayılar ve n>2 olmak koşuluyla, x^n + y^n = z^n denkleminin çözümü yoktur. Ben bunun kanıtını buldum, ama kanıtı bu kenar boşluğuna sığdırmak olanaksız.
Ancak bu kanıt bulunamamıştır. Fermat'tan sonra matematikçiler bu önermenin bir türlü içinden çıkamamışlardır. Fermat'ın bıraktığı defterler arasında teoremin kanıtına rastlayamadıkları gibi, kendileri de ne doğruluğunu ne yanlışlığını kanıtlayabilmişlerdir. Yıllar boyunca (300 yıl sonrasına kadar) bu konuda yapılan çalışmalar sonucu bu teoremin Shimura-Taniyama Konjektürü'nün bir özel durumu olduğu anlaşılmış, ardından da 1993'te İngiliz matematikçi Andrew Wiles, eski öğrencilerinden Richard Taylor'ın da yardımıyla ve cebirsel geometrinin çok karmaşık araçlarını kullanarak teoremi kanıtlamanın bir yolunu bulmuş ve bu kanıtı 1995'te Annals of Mathematics adlı dergide yayımlamıştır. Shimura-Taniyama Konjektürü'nün böylelikle ispatlanması sonucu Fermat'nın Son Teoremi de 1995'te ispatlanmış oldu.
Asal sayılar üzerinde çok durmuştur. Onun bu konuda çeşitli teoremleri vardır. Örneğin,
4n + 1
şeklinde yazılan bir asal sayı p, yalnızca bir tek şekilde iki karenin toplamı olarak yazılabilir.
Mesela en ufak asal sayılar p:
5 = 12 + 22 ve 13 = 22 + 32
dir. Bu teoremi daha sonra Euler kanıtlamıştır.
Pierre Fermat's father was a wealthy leather merchantsecond consul of Beaumont- de- Lomagne. There is some dispute [14] about the date of Pierre's birth as given above, since it is possible that he had an elder brother (who had also been given the name Pierre) but who died young. Pierre had a brothertwo sisterswas 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 the 1620s. In Bordeaux he began his first serious mathematical researchesin 1629 he gave a copy of his restoration of Apollonius's Plane loci to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrandduring this time he produced important work on maximaminima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat.
From Bordeaux Fermat went to Orléanshe studied law at the University. He received a degree in civil lawhe purchased the offices of councillor at the parliament in Toulouse. So by 1631 Fermat was a lawyergovernment official in Toulousebecause of the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat.
For 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-Lomagnea nearby town of Castres. From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he 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 senioritythe plague struck the region in the early 1650s meaning that many of the older men died. Fermat himself was struck down by the plaguein 1653 his death was wrongly reported, then corrected:-
I informed you earlier of the death of Fermat. He is alive,we no longer fear for his health, even though we had counted him among the dead a short time ago.
The following report, made to Colbert the leading figure in France at the time, has a ring of truth:-
Fermat, a man of great erudition, has contact with men of learning everywhere. But he is rather preoccupied, he does not report cases wellis confused.
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 mathematicsFermat told Carcavi about his mathematical discoveries.
In 1636 Carcavi went to Paris as royal librarianmade contact with Mersennehis group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies,he wrote to Fermat. Fermat replied on 26 April 1636 and, 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 spiralshis restoration of Apollonius's Plane loci. His work on spirals had been motivated by considering the path of free falling bodieshe had used methods generalised from Archimedes' work On spirals to compute areas under the spirals. In addition Fermat wrote:-
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 wishdo so without any ambition, from which I am more exemptmore distant than any man in the world.
It is somewhat ironical that this initial contact with Fermatthe 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 mathematiciansthis was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained.
RobervalMersenne found that Fermat's problems in this first,subsequent, letters were extremely difficultusually not soluble using current techniques. They asked him to divulge his methodsFermat sent Method for determining MaximaMinimaTangents to Curved Lines, his restored text of Apollonius's Plane locihis algebraic approach to geometry Introduction to PlaneSolid Loci to 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 maximaminima to his major work Cursus mathematicus. The widening correspondence between Fermatother 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' La Dioptrique by Beaugrand, Fermat paid it little attention since he was in the middle of a correspondence with RobervalÉtienne Pascal over methods of integrationusing them to find centres of gravity. Mersenne asked him to give an opinion on La Dioptrique which Fermat did, describing it as
groping about in the shadows.
He 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, minimatangents as reducing the importance of his own work La Géométrie which Descartes was most proud ofwhich he sought to show that his Discours de la méthode alone could give.
Descartes attacked Fermat's method of maxima, minimatangents. RobervalÉtienne Pascal became involved in the argumenteventually so did Desargues who Descartes asked to act as a referee. Fermat proved correcteventually Descartes admitted this writing:-
... 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 goodthat, if you had explained it in this manner at the outset, I would have not contradicted it at all.
Did this end the matterincrease 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 cycloid (which is indeed correct), Descartes wrote to Mersenne claiming that it was incorrectsaying that Fermat was inadequate as a mathematiciana thinker. Descartes was importantrespectedthus was able to severely damage Fermat's reputation.
The period from 1643 to 1654 was 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 placefrom 1648 Toulouse was greatly affected. Finally there was the plague of 1651 which must have had great consequences both on life in Toulouseof 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 that
xn + yn = zn
has no non-zero integer solutions for x, yz when n > 2. Fermat wrote, in the margin of Bachet's translation of Diophantus's Arithmetica
I have discovered a truly remarkable proof which this margin is too small to contain.
These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670.
It 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 June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. In November 1994 Wiles again claimed to have a correct proof which has now been accepted.
Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theorya wealth of other mathematical discoveries.
Fermat's correspondence with the Paris mathematicians restarted in 1654 when 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,was well aware of Fermat's outstanding mathematical abilities. Their short correspondence set up the theory of probabilityfrom this they are now regarded as joint founders of the subject. Fermat however, feeling his isolationstill 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:-
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 conciserelieve me of a burden that my duties prevent me from taking on.
However Pascal was certainly not going to edit Fermat's workafter 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:-
Two mathematical problems posed as insoluble to French, English, Dutchall mathematicians of Europe by Monsieur de Fermat, Councillor of the King in the Parliament of Toulouse.
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 of Nx2 + 1 = y2 for N not a square, was however solved by WallisBrounckerthey 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 cube (a 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 of x2 + 4 = y3that the equation x2 + 2 = y3 has 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 publicationhe turned to Fermat for help with the Fermat - Descartes correspondence. This led Fermat to look again at the arguments he had used 20 years beforehe looked again at his objections to Descartes' optics. In particular he had been unhappy with Descartes' description of refraction of lighthe now settled on a principle which did in fact yield the sine law of refraction that SnellDescartes 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.
In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probabilitythe correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hardin New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others.
Fermat described his method of infinite descentgave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. For suppose some number of the form 4k + 1 could not be written as the sum of two squares. Then there is a smaller number of the form 4k + 1 which 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 in [9] as
Secretivetaciturn, he did notto talk about himselfwas 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] timethat [France] place.