Proofs from THE BOOK by Martin Aigner and Günter Ziegler begins by giving six proofs of the infinity of primes. We will go over the third proof. Before we go over this proof, lets cover some background.
— 1. Fermat Numbers —
Fermat numbers are defined by
so that and . They are of great interest in many ways: for example, it was proved by Gauss that, if , is a prime , then a regular polygon of sides can be inscribed in a circle by Euclidean methods. The property of the Fermat numbers which is relevant here is
We will prove this theorem later.
The first four Fermat numbers are prime, and Fermat conjectured that all were prime. Euler, however, found in 1732 that
In 1880 Landry proved that
It is currently known that , is composite for . Factoring Fermat numbers is extremely difficult as a result of their large size. has known factors with remaining (where denotes a composite number with digits). has known factors with remaining. has no known factors but is composite. There are currently four Fermat numbers that are known to be composite, but for which no single factor is known: and . In all the other cases proved to be composite a factor is known. No prime has been found beyond , and it seems unlikely that any more will be found using current computational methods and hardware.
— 2. Infinitude of Primes Theorem —
We are now ready to prove Euclid’s Second Theorem, also called the Infinitude of Primes Theorem using the third proof in Proofs from THE BOOK.
from which our assertion follows immediately. Indeed, if is a divisor of, say, and , then divides , and hence or . But is impossible since all Fermat numbers are odd.
To prove the recursion we use induction on . For we have and . With induction we now conclude