Why Dunham's Focus On Number Theory And Fermat

While one can easily come up with many straight forward applications of analytic geometry, finding the slope of functions for maxima and minima, and optics, a little less straight forward may be the applications of his work in number theory. However, it is agreed upon by most that Fermat's most impressive contributions come from the field of Number Theory (Dunham 223). So does this mean that all of his work are just interesting facts about mathematics with no applications? Fermat's focus on the prime numbers have lead to results that are still in use today. The prime numbers possess a very interesting quality, they occur at random (or at least we have not discovered a way to find them) and there is no way to easily factor them. This is the foundation of the RSA algorithm used for encryption. This makes use of the fact that you can multiply two large primes together very easily, however factoring it will take a very long time. Fermat's little theorem is relied on as the decryption method for exponential encryption with a prime modulus. Hence all of Fermat's work on prime is used heavily for encryption methods. Still, this is just one of many applications of his work.

While Fermat never provided us with the proofs of his work, William Dunham in "Journey through Genius" describes briefly some of the work in number theory that Fermat did as well as the proofs done by Euler to prove Fermat's little theorem. The proofs use nothing more than basic algebra and are fairly short.
Euler begins with three proofs that Dunham describes before he works to proving the Fermat's little theorem. All theorems build on the ones proven previously. Finally Theorem 4 is the proof of Fermat's little theorem (Dunham 229). The proofs go as follows:

Dunham also describes Fermat's conjecture about prime numbers called "Fermat numbers" being disproven by Euler through his advanced methods of factoring numbers. (Dunham 229)

Dunham, William. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, 1990. Print.