Pierre de Fermat

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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 theory and a 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, 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:-

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.

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:-

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.

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  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 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 = y3 and that 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 publication and he turned to Fermat for help with the Fermat -  Descartes correspondence. This led Fermat to look again at the arguments he had used 20 years 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.

In 1656 Fermat 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 in 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 descent and gave 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 as

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.

Carl B Boyer, writing in, says:-

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.

J J O'Connor and E F Robertson