Term Papers on Pierre De Fermat

Term Paper Title	Pierre De Fermat
# of Words	1178
# of Pages (250 words per page double spaced)	4.71

Pierre de Fermat

     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 Pascal’s interest in a Parisian gambler’s 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 Fermat’s 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).  Fermat’s first theorem concerning primes was known as “Fermat’s 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 Girard’s 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 sq...

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