Pierre de Fermat

Pierre de Fermat was born on August 17, 1601, in Beaumont-de-Lomagne, a small bastide town in Gascony in southwestern France ^[1]. His father, Dominique Fermat, was a prosperous leather merchant who also served as one of the town's consuls, and the family's comfortable standing opened doors that a purely rural upbringing would have kept shut ^[2]. The house where the family lived still stands in Beaumont-de-Lomagne and now operates as a museum devoted to its most famous resident ^[3].

Details of his schooling are thin, a recurring problem for anyone assembling a Pierre de Fermat biography, since he left almost no personal papers describing his youth. Historians generally place his early education with the Franciscans in his home town, followed by studies at the University of Toulouse ^[2]. What is certain is that he emerged with an exceptional command of languages. He read Latin and Greek fluently, worked comfortably in Italian and Spanish, and later composed verse in several of these tongues ^[1]. That philological training mattered: his mathematics grew directly out of close reading of ancient Greek authors.

In the late 1620s he moved to Bordeaux, where he began his first serious mathematical work. There he circulated a restoration of a lost treatise by Apollonius of Perga on plane loci and started extending the maximum and minimum methods that would soon make his name among French mathematicians ^[1]. He then took a law degree at the University of Orleans in 1631, the credential that fixed the course of his professional life ^[2].

In 1631 Fermat purchased the offices of councillor and commissioner of requests at the Parlement of Toulouse, the regional high court, a common route to office in seventeenth century France where judicial positions were bought and could be inherited ^[1]. The purchase entitled him to add the noble particle to his name, and from then on Pierre Fermat signed himself Pierre de Fermat. The same year he married Louise de Long, a cousin of his mother, and settled into the life of a provincial magistrate ^[2].

His legal career advanced steadily. He moved through the chambers of the parlement, served in the criminal court, and in 1638 rose to a higher chamber ^[1]. In 1648 he acted for a period in the Chambre de l'Edit at Castres, the special tribunal that heard disputes between Catholics and Protestants under the Edict of Nantes ^[2]. Contemporaries left mixed assessments of him as a judge: conscientious and learned, though some complained he was distracted, which is not hard to believe given what occupied his evenings.

Mathematics entered the wider world through correspondence. From 1636 he exchanged letters with Marin Mersenne, the Minim friar in Paris whose cell functioned as the clearinghouse of European science, and through Mersenne with Descartes, Roberval, and others ^[4]. Fermat published almost nothing under his own name during his lifetime. He announced results in letters, often withholding proofs and challenging his correspondents to find them, a habit that won him admirers and irritated rivals in roughly equal measure ^[1].

Anyone asking who was Pierre de Fermat usually meets the answer through number theory, the field he effectively founded in its modern form. Reading a Latin edition of the Arithmetica of Diophantus, he filled its margins with observations about whole numbers ^[4]. His little theorem states that if p is a prime number and a is any integer not divisible by p, then a raised to the power p minus 1 leaves remainder 1 when divided by p. That result now sits at the base of modern cryptography, including the RSA algorithm that secures internet transactions ^[5]. He also studied the primes now called Fermat primes, characterized which numbers can be written as sums of two squares, and developed the method of infinite descent for proving statements about integers ^[1].

The most famous marginal note asserted that no three positive integers can satisfy the equation x^n + y^n = z^n for any integer exponent n greater than 2. Fermat wrote that he had a marvelous demonstration that the margin was too narrow to contain. The claim, known as Fermat's Last Theorem, defeated every mathematician who attempted it until Andrew Wiles completed a proof in 1994, published in 1995, using tools invented centuries after Fermat's death ^[6]. Most historians doubt Fermat possessed a valid general proof, though he did prove the case n equals 4.

His achievements ran well beyond arithmetic. In a manuscript circulated around 1636, before Descartes published La Geometrie, Fermat described how equations in two unknowns define curves, making him an independent co-founder of analytic geometry ^[1]. His method of adequality for finding maxima, minima, and tangents anticipated the differential calculus, and Newton later acknowledged drawing on Fermat's tangent methods ^[4]. In physics, his principle of least time explained the refraction of light by proposing that a ray travels the path requiring the shortest time, an idea that grew into the variational principles at the heart of modern physics ^[5]. And in a celebrated 1654 correspondence with Blaise Pascal about how to divide the stakes of an interrupted game, the two men laid the foundations of probability theory ^[7].

Fermat's marriage to Louise de Long in 1631 produced five children who survived to adulthood: two sons and three daughters ^[2]. The elder son, Clement-Samuel, followed his father into the law and became his literary executor. Two of the daughters entered religious orders, an ordinary path for daughters of the provincial magistracy in that era ^[2].

By the standards of the great scientific egos of his century, Fermat lived quietly. He never traveled to Paris, never sought a court position, and turned down repeated urgings from friends to publish his results properly ^[1]. His disputes stayed on paper. The sharpest was with Descartes, who in 1638 attacked Fermat's method of tangents after Fermat had criticized Descartes's account of refraction. The quarrel ran hot for months before Descartes conceded, grudgingly, that the tangent method was sound ^[4].

His classical learning shaped his private hours as much as mathematics did. He annotated Greek texts, corresponded on philological questions, and was consulted on emendations to ancient authors ^[1]. This bookish streak explains the form his mathematics took: commentary, restoration, and marginal notes on the ancients, rather than systematic treatises of his own.

The 1650s tested him. Plague swept through southern France in 1652 and early 1653, and Fermat fell so gravely ill that a colleague reported his death to correspondents in Paris; the report proved premature, and he recovered to serve nearly twelve more years on the bench ^[2]. His judicial duties grew heavier in this final period even as his mathematical energy stayed high. The correspondence with Pascal on probability came in 1654, and his work on the theory of numbers continued into the 1660s ^[7].

In 1657 he issued a challenge to the mathematicians of Europe concerning the equation now called Pell's equation, asking for whole number solutions. English mathematicians John Wallis and William Brouncker responded, and the exchange, conducted with some transparent national rivalry, pushed the study of such equations forward ^[1]. Fermat also corresponded with Christiaan Huygens late in life, attempting with limited success to interest the younger generation in his arithmetic discoveries ^[4].

He died on January 12, 1665, in Castres, three days after presiding over a court session there, and was buried in Castres before his remains were later moved to the family vault in Toulouse ^[2]. In 1670 his son Clement-Samuel published a new edition of Diophantus containing his father's marginal observations, and in 1679 a fuller collection of his papers appeared as Varia Opera Mathematica. Without those posthumous volumes, much of his work would simply have vanished ^[1].

Fermat's reputation grew after his death in a way few careers built on unpublished letters ever have. Leonhard Euler in the eighteenth century took up the number theoretic claims one by one, proving most and refuting one (Fermat's conjecture that all numbers of the form 2 raised to a power of 2, plus 1, are prime fails at the fifth case, as Euler showed in 1732) ^[5]. Gauss, Kummer, and generations after them built number theory on questions Fermat first posed. Among Pierre de Fermat achievements, this posthumous agenda setting may be the largest: he defined problems that occupied mathematics for three centuries.

The Last Theorem became the most famous unsolved problem in mathematics, generating vast amounts of new theory in failed attempts before Wiles, building on work of Ken Ribet, Gerhard Frey, and the Taniyama-Shimura conjecture, finally settled it in the 1990s ^[6]. Wiles received the Abel Prize in 2016 for the proof. The story turned a seventeenth century margin note into a piece of popular culture, the subject of bestselling books and television documentaries.

Among basic Pierre de Fermat facts worth keeping in view: he was never a professional mathematician, held no academic chair, and published almost nothing while alive, yet Fermat's little theorem runs inside the encryption protecting everyday digital life, and his least time principle survives in optics courses everywhere ^[5]. His statue stands in Beaumont-de-Lomagne, a Toulouse street and a lycee carry his name, and historians still rank the amateur magistrate of Toulouse among the greatest mathematicians of any century ^[3].