Table of contents |
2 Primality of Fermat numbers 3 Factorisation of Fermat numbers 4 Relationship to Constructible Polygons |
The Fermat numbers satisfy the following recurrence relations
for n ≥ 2. From the last equation, it follows that no two Fermat numbers share a common factor. To see this, suppose that 0 ≤ i < j and F_{i} and F_{j} have a common factor a > 1. Then a divides both
Here are some other basic properties of the Fermat numbers:
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the first five Fermat numbers F_{0},...,F_{4} are easily shown to be prime. However, this conjecture was put to rest by Leonhard Euler in 1732 when he showed that
It is widely believed that Fermat was aware of Euler's result, so it seems curious why he failed to follow through on the straightforward calculation to find the factor. One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work.
There are no other known Fermat primes F_{n} with n > 4. It is not known whether these are the only Fermat primes, and it is not even known whether or not there are infinitely many Fermat primes.
The main result which is helpful in determining the primality of Fermat numbers is the following test, due to Pepin:
Pepin's test -- Let F_{n} be a Fermat number. Then F_{n} is prime if and only if
Since at least the time of Euclid, it was known that many regular polygons were capable of being constructed by ruler and compass. The problem then arose, "For which positive integers n is the regular n-gon constructible with ruler and compass?". Carl Friedrich Gauss was the first to make progress on this problem by showing its relationship to Fermat primes. In 1796 (at the age of 19), Gauss showed that a sufficient condition for the regular n-gon to be constructible is that n is a power of 2 or the product of a power of 2 and distinct Fermat primes. In other words, if n is of the form n = 2^{k}p_{1}p_{2}...p_{s}, where k is a nonnegative integer and the p_{i} are distinct Fermat primes, then the regular n-gon is constructible. The necessity of this condition was not proved until 1836 by Pierre Wantzel. A positive integer n is of the above form if and only if φ(n) is a power of 2, where φ(n) is Euler's totient function.
See also: