It is not often a person contributes to a field they do not even work in the way Pierre de Fermat has contributed to the field of mathematics. Born to a wealthy leather merchant, Fermat received a bachelor’s in civil law from the University of Orléans and went on to become a lawyer, while at the same time engraving his name into math history books, doing said math just for recreation. His importance in mathematics lead to many theorems named after him, as well as numbers. These numbers are known as Fermat numbers, which are positive integers, that take of the form F_{n} = 2^{(2n)} + 1, when n is nonnegative and an integer. For example for F_{1}, F_{1}= 2^{(2)}+1= 5. The first five Fermat numbers are 3, 5, 17, 257, and 65537, and these numbers continue to grow to incredibly large magnitudes. Fermat believed this form created an infinite number of prime numbers, which are known as Fermat primes.

Fermat numbers are occasionally written as 2^{n}+1, but since when n is greater than zero and F_{n} prime, n must be a power of two, the form F_{n} = 2^{(2n)} + 1 is the common form for Fermat numbers. One of the main problems with Fermat claiming all these numbers are prime is the fact that they soon become too large to calculate for even today’s computers, let alone a man with his pen and paper in the 17^{th} century. Unfortunately for Fermat, by the time the 18^{th} century rolled around, he was dead. In 1732, mathematician Leonhard Euler found that F_{5}, which is 4,294,967,297, is actually divisible by 641, most likely figuring this out from having a large amount of time on his hands. While this showed that some Fermat numbers are not actually prime, excluding when n=0 in the form 2^{n}+1, it does not discount the fact that the Fermat number equation could still make an infinite number of primes, since there are infinite amount of Fermat numbers. However as of now, the only Fermat primes that are known are F_{0} through F_{4}.

Now initially I found the idea of an equation, the equation here being F_{n} = 2^{(2n)} + 1, that finds only certain prime numbers, most of which are way too large to even be calculated even 400 years after the equation for them was created, the equivalent to a student doing extra credit when he has a 98% in his class. What I’m trying to say is, I found Fermat numbers pointless and to be the 17^{th} century mathematician’s version of a braggadocio. However, I know nothing and Gauss managed to find a relationship between “Euclidean construction of regular polygons and Fermat primes,” where he showed a regular 17-gon could be constructed. It was also found a regular n-gon can be created if n is the product of any number of Fermat primes and the number 2. These regular n-gons take the property of being able to be constructed with a compass and straightedge. Who would have thought one of the greatest mathematicians to ever live could leave me feeling so inadequate, at least mathematically.

Similar to Fermat numbers are what are known Mersenne numbers, created by 17^{th} century French mathematician and music theorist Marin Mersenne. Yet again a person being a jack of all trades, except instead of being a master of none they were a master of a few or at least of one. Mersenne numbers take of the form M_{n} = 2^{n} -1, and Mersenne primes are numbers that take that form which are prime. Mersenne believed that for n<=257, M_{n} was prime for n= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and the rest are composite. While this belief turned out to incorrect, he still got the name for the primes. Just like Fermat primes, it is unknown whether there are an infinite number of Mersenne primes, but as of now 48 Mersenne primes are known, the largest being 2^{57885161}-1, which again makes me wonder how much time do some of these mathematicians have on their hands.

Mersenne numbers were originally studied because of their connection to perfect numbers, which are positive integers that are equal to the sum of their divisors. Euclid proved that if the number 2^{n}-1 is prime, then 2^{n-1}(2^{n}-1) is a perfect number, which many years later led to Euler discovering that all even perfect numbers come in this form. Another interesting fact is that the ten largest known prime numbers are Mersenne numbers. I personally find number theory incredibly interesting, partly because I like numbers and partly because how mathematicians are able to come with these theorems and proofs baffle me. I ultimately wonder if they had any true goals when thinking about these primes, or if it was just for the pure fun and interest in it.

References:

http://en.wikipedia.org/wiki/Fermat_number

http://interact.sagemath.org/edu/2010/414/projects/tsang.pdf

http://mathworld.wolfram.com/FermatNumber.html

http://mathworld.wolfram.com/FermatPrime.html

https://primes.utm.edu/glossary/page.php?sort=FermatNumber

http://en.wikipedia.org/wiki/Mersenne_prime

http://en.wikipedia.org/wiki/Marin_Mersenne

http://en.wikipedia.org/wiki/Pierre_de_Fermat

http://mathworld.wolfram.com/MersennePrime.html