Pierre de Fermat Ismail Tawil Fermat's life Pierre de Fermat, a French mathematician born on the 17th of August, 1601 in Beaumont de Lomagne, France and passed away on the 12th of January, 1665 in Castres, France. He was the son of a successful leather merchant. While not much is known of his early life and education, there is evidence that his primary education might have been at a local Franciscan monastery; it is known that Fermat did attended the University of Toulouse and then later moved to Bordeaux in the second half of the 1620s. Fermat was fluent in Greek, Latin, Spanish, and Italian. He also studied a little in philological problems and even some Latin poetry. Fermat began his first critical mathematical research in Bordeaux. From Bordeaux Fermat went to the University of Orlans, where he studied law. He later received a Bachelor's Degree in Civil Law. His mother and her side of the family were Pierre biggest influence for studying law and for deciding to work in Parliament. Fermat eventually married and had five children. By 1631, Fermat was a lawyer and a government official in Toulouse, as well as given the title of "commissioner of requests." In 1648, Fermat was promoted to a king's councilor-ship in the parliament of Toulouse. The stature he now held entitled him to change his name from Pierre Fermat to Pierre de Fermat ("de" is the mark of nobility in France). From his appointment in 1631, Fermat worked in the lower chamber of parliament but in 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level in the criminal court. Promotion was done mostly on seniority and when the plague struck the region in the early 1650s, many of the older men died. Fermat himself was afflicted with the plague and in 1653 his death was wrongly reported and was later corrected. For the most part, math was a hobby to Fermat seeing as how he was a busy lawyer and did not let his love of math take up all of his time; however, he was at times preoccupied with his mathematical progress, so much so that he continued to cherish his mathematical friendship with a mathematician named Beaugrand, who he had previously done research with, even after he moved to Toulouse. Fermat also met another mathematician named Carcavi during his work in the government due to the fact that both of them were councilors in Toulouse; they both had a shared love for mathematics where Fermat discussed with Carcavi about his mathematical researches ad well as discoveries. During his research in Bordeaux, Fermat studied the works of a French mathematician from the 16th century, named Franois Vite. Vites idea of algebra as an Analytic Art largely guided Fermat in his choice of the type of problems he had became interested in. Fermat viewed his own work as a sort of continuation to the Vitan tradition. From Vite, Fermat inherited the idea of Symbolic Algebra as a tool made for uniting the realms of both Geometry and Arithmetic, which would later influence his work in Number Theory. Algebraic equations had meaning in both fields, depending on whether the variables were line segments or numbers. Moreover, Vites Theory of Equations shifted Fermat's attention away from the solutions of certain formulas to the relationships between the solutions and the structures of their parent equations or between the solutions of one equation and those of another. Most of Fermats research headed toward Reduction Analysis, where any given problem could be reduced to another or identified with a certain class of problems for which the general solution was known. This Reduction Analysis could be reversed, in most cases, to act as a generator of families of solutions to problems. Due to his love for language and literature and Vite's influence on his young brilliant mind, Fermat was very concerned with the restoration of a number of lost texts and so he began to restore lost works such as Apollonius Plane Loci, Euclids Porisms, and Diophantus Arithmetica. In terms of his mathematical work, there is speculation that Fermat never wanted any of his work to be published as he had considered math to be but a hobby. However, the few works that he did publish, he published them anonymously. Many of Fermat's work was done by mail or correspondence; he would mail many other famous mathematicians his progress in the mathematical field. Fermat rarely provide his proofs in his correspondence with other mathematicians; usually stating the theorems, however, he had always managed to neglected the proofs of these such theorems. Due to his "negligence", many of his theorems were left without any proofs, which eventually led to much doubt as to his authenticity as a mathematician. In fact, his most famous work, which was dubbed 'Fermat's Last Theorem', stayed without a proof until, in 1993, a mathematician named Andrew J. Wiles provided the first proof to this particular theorem. Fermat received very little recognition as a mathematician during his life, and if not for the fact that other mathematicians saved his papers and letters, he may not have had the legacy that he has to this day. Together with Ren Descartes, a very distinguished French mathematician in the 17th century, Fermat was one of the two leading mathematicians of the first half of the 17th century. Following his death, his oldest son, Clment-Samuel inherited his law offices in 1665 and then later decided to publish all of his fathers mathematical papers. Fermat's mathematical works The Method of Maxima and Minima An astronomer and mathematician by the name of Johannes Kepler, influenced the development of Fermat's early mathematical works. Kepler began to observe the volumes for specific dimensions. Keplers observations determined that as a maximum value was approached from either side, the change in volume started to become increasingly small. Bearing this same concept in mind, Fermat chose to expand on this wokr by finding maximum or minimum values of any of the given functions. Fermat understood that if a function f(x) had a maximum or minimum at x, then making e to be very small, it can be shown that the value of f(x − e) was almost equal to that of f(x). Therefore, this method involved what can be called a "pseudo-equality," in which Fermat carefully set: f(x) = f(x − e) However, before Fermat could find the roots of this equation, which would enable him to find the maximum and/or minimum of the function, he would need to correct the function's "pseudo-ness" by letting e "assume the value of zero". 2 Calculus and Curves During his work on the properties of curves, Fermat helped in the development of the mathematical field of calculus. As his studies of curves and their equations continued, Fermat noticed he could take the equation of an ordinary parabola (ay = x2), as well as an equation for a rectangular hyperbola (xy = a2), and generalize it to the follwowing form an−1y = xn The curves made by this equation are known as the parabolas/hyperbolas of Fermat (n is positive or negative). He similarly generalized the Archimedean spiral, r = aQ In the 1630s, he discovered an algorithm from these curves, that was similar to differentiation. This procedure let him find tangents of curves and locate the maximum, minimum, and inflection points of polynomials. For any equation, Fermat's way of finding the tangent at any given point was to find the subtangent for that specific point. Fermat found an equation for the areas bounded by curves through a summation process; this process has now been dubbed Integral Calculus. Such an equation is: Z A = xn dx = an+1=(n + 1) It is not known whether or not Fermat noticed that the differentiation of xn, leading to nan−1 was actually the inverse of integrating xn. Fermat took on problems involving even more general algebraic curves. Fermat applied his knowledge of infinitesimal quantities to many other problems, including the calculation of centers of gravity as well as finding the length of a curve. Fermat did not notice the idea of what is now known as the Fundamental Theorem of Calculus, however, his work in this area aided in the development of Differential Calculus. Additionally, he contributed to the Law of Refraction1 by disagreeing with his colleague, a philoso- pher and amateur mathematician, Ren Descartes. Fermat had claimed that Descartes had incorrectly deduced his Law of Refraction because it was deep-seated in his assumptions. As a result, Descartes became irritated and attacked Fermat's Method of Maxima and Minima. Fermat differed with Carte- sian views concerning the Law of Refraction, which was published by Descartes in 1637. Descartes attempted to justify the Sine Law through the assumption that light travels faster the denser the two media involved in the refraction. Twenty years later, Fermat noticed that this seemed to conflict with the view of the Aristotelians where nature always chooses the shortest path. Number Theory 1According to Fermat's theory, if light passes from point A to point B, while being reflected and refracted in any manner during the passage, the path it takes can be calculated and the time it takes to pass from A to B is an extreme 3 One may define a prime number as any natural number greater than 1 which has a divisor (without remainder) of only 1 and the number itself. Fermat noticed that numbers such as 3; 5; 17; 257; 65537 all belonged to a single sequence and can be generated by one simple process: 3 = 2 + 1; 5 = 22 + 1; 17 = 24 + 1; 257 = 28 + 1; 65537 = 216 + 1 The sequence seen here is such an equation 2n 1 2 + 1 jn=1 Now if we wanted to see if one of these numbers is prime it would be a difficult and lengthy process, unless we follow and understand Fermat who claimed that all numbers in the previous sequence are prime.