The Mathematics Of Fermat

Of course what is known most about Pierre de Fermat is his work in mathematics. Carl Boyer devotes a section to Pierre de Fermat in "A History Of Mathematics" which describes a great deal of his mathematical contributions. One of Fermat's mathematical endeavors was to redo the Plane Loci of Apollomius. In 1636 from this project the fundamental principle of analytic geometry came about. While Fermat and Descartes at this time both had different views on this new subject of analytic geometry, Fermat stayed with the symbolic notation and style of Viete. In fact, while Fermat was working with sketching equations, he discovered new approaches to solving problems. Additionally Fermat developed the proposition

"Every equation of first degree represents a straight line"

 Fermat spent time showing equations that produced, hyperbolas, parabolas, and ellipses. He reduced homogeneous equations into easier to work with forms and gave results of dealing with equations such as

which he knew produced a straight line and was only working within the first quadrant. This type of equation, for example, was a form he worked to reduce other quadratic equations to.

Fermat's "Introduction to Loci" was one of his works, but like all of his others except the work he published anonymously, he did not publish. At this time the lack of publication did not revel Fermat's knowledge of analytic geometry as directly as Descartes, and moreover he was not given credit to the discovery. It was not until later that Fermat's work was uncovered and later realized they had similar findings at the same time. Both Fermat and Descartes kept geometry in a maximum of three dimensions.

Fermat's work however did not just stop at analytic geometry. It was Laplace who said "Fermat, the true inventor of the differential calculus." Fermat discovered methods of finding tangent lines which allowed him to also find maxima and minima. Unlike modern day, Fermat's method stems from acknowledging a point and one a very small distance away to provide similar triangles. To set equal the ratio of heights to lengths. This essentially assumes the distance between the two points is "zero" resulting in equal slopes of tangent lines and making use of right triangle relations to the slope "rise (height) over run (length)".

This can easily be related through cross multiplication and minor algebra to the modern day version:

In addition to discovering this method of differentiation, Fermat worked on a technique with sums of series which allowed him to find the area under the curve.  However the concept of these ideas being connected as in calculus

Above all of his accomplishments, Fermat is probably most known for his results in the field of Number Theory. Fermat developed a proof technique called "method of infinite decent" which is considered to be the reverse of mathematical induction. Out of all the work in this field, Fermat became most intrigued by prime numbers.

In fact, Fermat believed he discovered a formula that produced prime numbers. This however was dis-proven by Euler by a counter example. Still, the numbers generated from this Fermat are called "Fermat numbers" and still have some value to them. Assume n is a non negative integer.  Fermat numbers are found through the formula:

By Fermat, every Fermat number is supposed to be prime. This is true for n values 1,2,3,4. Euler showed in 1732 that when n=5, 4294967297 is not prime. So while Fermat's hypothesis was not true, the Fermat numbers still have open questions such as if the Fermat numbers are composite for all n>4, if there are infinitely many Fermat primes, and if there are infinitely many composite Fermat numbers.

Fermat also had another theorem which came from his interest of prime numbers known as "Fermat's little theorem." The theorem states:
"If p is a prime number, then for any integer a, the number a ^p − a is an integer multiple of p." Also written as:

Similarly a direct result of this theorem arises "If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a ^p − 1 − 1 is an integer multiple of p." Likewise this statement can be written as:

As stated before, this is an example of a theorem that Fermat never provided a proof for. 

One of the theorems Fermat is most known for his the one referred to as "Fermat's last theorem." Fermat's last theorem looks very similar to the Pythagorean theorem which states for the sides of a right triangle:

Fermat's last theorem says that there is no positive integer n greater than 2 such that:

This theorem was found in the margin of Fermat's copy of Arithmetica. Fermat claimed that he did have a proof, but it was too large to fit in that margin. Whether he actually did have a proof remains unknown, however the problem remained open for over 300 years before Andrew Wiles found a proof in 1994. The problem was considered to be one of the worlds hardest math problems. 

Fermat also did work with mathematics in the field of optics. Through this work probably the most known result is "Fermat's principle" or "principle of least time" which states "the path taken between two points by a ray of light is the path that can be traversed in the least time."

Boyer, Carl B. A History of Mathematics. New York: Wiley, 1968. Print.