Pierre de Fermat was born on August 20, 1601 in southwestern France. The son of a wealthy leather merchant he soon developed a great love for numbers and mathematics. I chose to do this project about him because he is responsible for numerous theorems concerning probability, calculus, analytical geometry, and prime numbers. Furthermore, he is also responsible for the creation of one of the greatest mathematical riddles of all time. Probability, Calculus, and Analytical Geometry Fermat and Blaise Pascal established the probability theory still used today through their exchange of letters.
Driven by Pascals interest in a Parisian gamblers problem concerning a game of chance, the two were certain to devise a set of mathematical rules that would accurately describe the laws of chance. Together, they would begin to build a completely new aspect of mathematics, which is today known as probability. Fermat was also responsible for the establishment of calculus, long before Isaac Newton was even born. Calculus, which is the ability to calculate the rate of change of two quantities respectively, as Fermat developed it helped scientists better grasp the concepts of velocity and other quantities.
It was Fermat that established the theorems that made the works of Isaac Newton, the father of calculus, possible. In fact, until 1934 it was unknown that Newton had mooched Fermats theories to establish his law of gravity. Fermat was also responsible for the establishment of many basic rules of geometry. He showed the any equation in the form xy=k^2 or in the form (a^2)+(x^2)=ky^2 can be graphed as a hyperbola. He then showed that (a^2)+/-(x^2)=by is a parabola, that (a^2)-(x^2)=ky^2 is an ellipse, and that (x^2)+(y^2)+2ax+2by=c^2 is a circle.
Prime Numbers Fermat also had a unique fascination with prime numbers. He had devised several theorems concerning the investigation of prime numbers. A prime number is any number whose factors are only itself and one (for example: 5 is a prime number since it is only divisible by 1 and 5). Fermats first theorem concerning primes was known as Fermats lesser theorem. In it, Fermat stated that integers in the form (2^2^n) +1 are always prime. He made this assumption based on the induction of only five cases in which it worked (n=0, 1, 2, 3, 4).
Ex: if n=1 then (2^2^1)+1 (2^2)+1 4-1= 3 3 is a prime number This Theorem was proved wrong a century later by another mathematician, known as Leonard Euler, who saw that this is false for numbers where n is equal to any integer between five and sixteen. Therefore, it is now pondered if there is even the existence of any other prime numbers that can be written this way beyond those that Fermat knew. Ex: if n=5 then (2^2^5)+1 (2^32)+1 4294967296+1=4294967297 Therefore 4294967297 is a prime number according to Fermat
However is not (as proved by Euler). Another theory concerning prime numbers that Fermat devised was based on the work of Girard. This mathematician asserted that all prime numbers of the form 4n+1 could be written as the sum of two squares. It can also be proven that 4n-1 is never the sum of two squares, and all prime numbers are either in the form 4n+1 or 4n-1. Fermat took Girards work one step further and asserted that all primes could easily be classified into those that are and are not the sum of squares.
Fermat also knew that all primes could be expressed as the difference between two squares in one unique way. Fermats Last Theorem It was in the margin of the Artihmetica, a ancient text written (in Latin) by Diaphantus (a third century mathematician), that Fermat would write one of the greatest mathematics riddles of all time. As he was reading about Pythagorean triples and the breaking down of a square into the sum of two other squares, Fermat noted this in the margin:
Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere. Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet. Which translates into: It is impossible for a cube to be written as a sum of two cubes or for a biquadrate to be written as the sum of two biquadrates, or generally any power which is greater then the second to be written as the sum of two like powers.
I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain. Fermat, however, never wrote the proof anywhere else in his notes, and so began the worlds quest to attempt to solve this riddle. In two sentences Fermat had created a problem that would plaque mankind for centuries. He had basically stated that even though the Pythagorean theorem, (a^2)+(b^2)=c^2, is valid in many cases, if the exponent is raised to equal any number greater then 2, (a^n)+(b^n)=c^n when n>2, the equation could never be true.
Hundreds of mathematicians have jumped at the opportunity to try to prove Fermats last theorem, as it was called. The greatest mathematicians from the span of the past three centuries have tried and failed in proving this theory. Everyone from Euler to Sophie Germain has given it a shot and in 1908 a mathematician by the name of Paul Wolfskehl offered a $2,000,000 reward to anyone who could solve the theorem. All efforts were failures until Andrew Wiles, a 41-year-old mathematician from Princeton University, finally succeeded.
Drawing on the aspects of the Shimura-Taniyama conjuncture for subset elliptic curves (described through the cubic equation of y^2=(x^3)+a(x^2)+bx+c), Wiles was able to step out with his hundred page proof in 1993. However, he soon retracted his proof due to some minute theoretical problems that needed to be patched. Then, 14 months later, he returned and finally published his valid proof on October 25, 1994. And so, the achievements of the Pierre de Fermat, no doubly one of the greatest mathematicians of all time, were great.
His knowledge and expertise in so many areas of mathematics is unbelievable, yet he is best known for his last theorem. In my opinion, the solution to the last theorem was never found by Fermat himself. After all, Fermat only published his last theorem days before his death, with no time to explain or even give any inkling describing his proof. Whether or not its discoverer knew the solution to Fermats last theorem is unknown and will always be. The solution, however, is now known, following hundreds of years of curiosity and anxiety to solve the greatest mathematical riddle left behind by one of the greatest mathematical minds.