The * Fermat numbers* are numbers of the form

Fermat thought that all the were prime. The first five are:

However, it turns out that . Note that

Therefore,

Here x is an integer.

On the other hand, , so

This proves that .

A * Fermat prime* is a Fermat number which is
prime. It is an open question as to whether there are infinitely many
Fermat primes.

Surprisingly, Fermat primes arise in deciding whether a regular n-gon (a convex polygon with n equal sides) can be constructed with a compass and a straightedge. Gauss showed that a regular n-gon is contructible with a compass and a straightedge if and only if n is a power of 2 times a product of distinct Fermat primes.

Here are some properties of the Fermat numbers.

* Proposition.* If p is prime and , then for some
k.

I won't prove this result, since the proof requires results about quadratic residues which I won't discuss for a while. Here's how it can be used.

* Example.* Check
for primality.

Here , so all prime divisors must have the form . There are around 1024 numbers less than 65537 of this form, but I only need to check numbers up to the square root . (For if a number has a prime factor, it must have a prime factor less than its square root.)

(The next value of is 257, which is larger than .) Conclusion: 65537 must be prime!

* Proposition.* for .

* Proof.* and , so . The result is true for
.

Take , and assume the result is true for n; I'll try to prove it for . By assumption,

Now

That is,

This is the statement for , so the proof is complete, by induction.

* Proposition.* If , .

* Proof.* Assume (if not, switch m
and n). Suppose p is prime and and . Also, since
implies that occurs in this
product. But

Since p divides both of the terms on the right, , so . This is impossible, since all the 's are odd.

Therefore, there is no prime dividing both and , and hence .

Copyright 2019 by Bruce Ikenaga