| Euclid may have been the first to give
a proof that there are infintely many primes, but his proof has been followed
by many others. Below we give Goldbach's clever proof using the Fermat numbers
(written in a letter to Euler, July 1730), plus a few variations. See the
page "There are Infintely Many Primes" for several more
First we need a lemma.
Now we can prove the theorem:
- The Fermat numbers Fn = are pairwise
- It is easy to show by induction that
F0.F1.....Fm-1. This means that if d divides both Fn and
Fm (with n < m), then d
also divides Fm-2; so d divides 2. But
every Fermat number is odd, so d is one.
- There are infinitely many primes.
Choose a prime divisor pn of each
Fermat number Fn. By the lemma we know these primes
are all distinct, showing there are infinitly many primes.
Note that any sequence that is pairwise relatively prime will
work in this proof. This type of sequence is easy to construct. For example, choose relatively prime
integers a and b, then define an as follows.
This includes both the Fermat numbers (with a=1,b=2)
and Sylvester's sequence (with a=1,b=2):
a1=2 and an+1=
In fact, the proof really only needs a sequence which has a subsequence which is
pairwise relatively prime, e.g., the Mersenne numbers.