Cindy Tsang University of Washington Math414 Number Theorey Professor William Stein March 12, 2010
1 Introduction 2 Background of Fermat Numbers 3 Geometric Interpretation of Fermat Numbers 4 Factoring Status of Fermat Numbers 5 Basic Properties of Fermat Numbers 6 Primality of Fermat Numbers 7 Mersenne Numbers and Fermat Numbers 8Infinitude of Fermat Primes 9 Divisibility of Fermat Numbers 10 References 2 3 5 6 7 12 17 19 21 23
Prime numbers are widely studied in the field of number theory. One approach to investigate prime numbers is to study numbers of a certain form. For example, it has been proven that there are infinitely many primes in the form a + nd, where d ≥ 2 and gcd(d, a) = 1 (Dirichlet’stheorem). On the other hand, it is still an open question to whether there are infinitely many primes of the form n2 + 1. In this paper, we will discuss in particular numbers of the form 2 + 1 where n is a nonnegative integer. They are called Fermat numbers, named after the French mathematician Pierre de Fermat (1601 – 1665) who first studied numbers in this form. It is still an open problem towhether there are infinitely many primes in the form of 2 + 1. We will not be able to answer this question in this paper, but we will prove some basic properties of Fermat numbers and discuss their primality and divisibility. We will also briefly mention numbers of the form 2n – 1 where n is a positive integer. They are called Mersenne numbers, named after the French mathematician Marin Mersenne. Insection6, we will see how Mersenne numbers relate to the primality of Fermat numbers.
Pierre de Fermat (1601 – 1665)
Marin Mersenne (1588 – 1648)
[pictures from http://en.wikipedia.org/Pierre_de_Fermat & http://en.wikipedia.org/wiki/Marin_Mersenne]
2 Background of Fermat Numbers1
Fermat first conjectured that all the numbers in the form of 2 + 1 are primes. However, in 1732,Leonhard Euler refuted this claim by showing that F5 = 232 + 1 = 4,294,967,297 = 641 x 6,700,417 is a composite. It then became a question to whether there are infinitely many primes in the form of 2 + 1. Primes in this form are called Fermat primes. Up-to-date there are only five known Fermat primes. (See section4 for more details on the current status of Fermat numbers.)
Leonhard Paul Euler(1707 – 1783)
Carl Friedrich Gauss (1977 – 1855)
[pictures from http://en.wikipedia.org/wiki/Euler & http://en.wikipedia.org/wiki/Gauss]
In 1796, the German mathematician Carl Friedrich Gauss (1977 – 1855) found an interesting relationship between the Euclidean construction (i.e. by ruler and compass) of regular polygons and Fermat primes. His theorem is known as Gauss’s Theorem. Gauss’sTheorem2. There exists an Euclidean construction of the regular n-gon if and only if n = 2ip1p2···pj, where n ≥ 3, i ≥ 0, j ≥ 0, and p1, p2,…, pj are distinct Fermat primes.
All historical information in this section is from Reference1 Chapter1. A proof of Gauss’s Theorem can be found in Reference1 Chapter16.
Gauss’stheorem implies that all 2n-gons for n ≥ 2 are constructible. Moreover, since so far only five Fermat numbers are known to be prime, it implies that for n odd, there are only
+ 5 C1 + 5C1 + 5C1 + 5C1 = 31 n-gons that are known to be Euclidean constructible. If it turns
out that there is only a finite number of Fermat primes, then this theorem would imply that there is only a finitenumber of Euclidean constructible n-gons for n odd. The figure below shows five Euclidean constructible n-gons.
Triangle, pentagon, heptadecagon, 257-gon and 65537-gon.
[figure from Reference1 Chapter4]
3 Geometric Interpretation of Fermat Numbers
As Gauss’s theorem suggests, Fermat numbers might be closely related to some of the problems in Geometry. It is hence useful if we can...
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