In this post, we present an alternative proof that there are infinitely many prime numbers in addition to the basic proof presented in the previous post Euclid’s Proof of the Infinitude of Primes. Historically, this alternative proof is due to Christian Goldbach (from a letter to Leonhard Euler in 1730). It is an elegant proof that is worthy to be considered as straight from “The Book” by Paul Erdos (at least “The Approximate Book”). Specifically it is one of the six proofs for the infinitude of primes found in .
This proof of infinitude of primes uses the notion of Fermat numbers. We present the proof in two parts.
For a proof of the infinitude of primes using topology, see The Infinitude of Primes – a Topological Proof.
Theorem 1. There exist infinitely many positive integers such that for every and with , the integers and are relatively prime (i.e. if divides both and , then ).
The existence of implies that there are infinitely many primes. Each has at least one prime factor. Then choose the smallest prime factor for . It is clear that for (otherwise and would not be relatively prime).
One way to get the integers in Theorem 1 is through the Fermat numbers.
A positive integer is a Fermat number if it is of the form for some non-negative integer . The following two points establish Theorem 1.
- Each Fermat number can be generated recursively by multiplying all Fermat numbers before it and then add 2. We have the following:
We can prove the above recursive relation by induction. Clearly (the case for ). Suppose that the recursive relation holds for where . Thus we have or . The following derives the recursive relation for the general case where :
We now show that any pair of Fermat numbers are relatively prime. Suppose that is a common divisor for and where . Then divides both and . Because of the recursive relation , divides 2. Since the Fermat number is an odd integer, cannot be 2. Thus .
Thus the Fermat numbers satisfy Theorem 1.
See here for a generalization of the proof in this post, without the Fermat number point of view.
- Aigner, M., Gunter, M.Z. Proofs from THE BOOK, third edition, Springer-Verlag, Berlin, 2004.