## Friday, 22 April 2016

### Proof that there are infinitely many quaternion primes

Sometimes proofs can be annoyingly difficult. I have been reading about quaternions in Elements of Number Theory and Exercise 8.4.2 asks you to prove that there are infinitely many quaternion primes without assuming that every natural number is the sum of four squares. I have been racking my brains over this one for a few days and finally I came up with a proof that I think works today.

What we know is that a quaternion "prime" is defined to have a norm which is prime in Z otherwise the norm will end up being the product of two quaternions of smaller norm. I was originally thinking that we should adopt the proof of showing that there are infinitely many primes in Z (see page 2 of the book) by assuming a finite number of quternion primes q1,q2,...,qk and then forming a product of their norms and then adding 1. This new natural number is then not divisible by any of the norms norm (q1), norm (q2), ..., norm(qk). But this hits snags straight away because you don't know if this new number is prime in Z and you don't know if it even relates to a quaternion.

After much thought, here is my proof.

Let us suppose to the contrary that there are only a finite number of quaternion primes q1,q2,...,qk (where k is some natural number). Now the norm of each of these quaternion primes must be a prime in Z, so norm (q1)=p1, norm (q2)=p2, ... ,norm(qk)=pk where p1,p2,...,pk are primes in Z.

Now there are an infinite number of primes in Z of the form 4n+1 (see page 113) and so we can always choose a prime p of this form which is not one of p1,p2,...,pk. Further by the two-square theorem (see page 109) p=a2+b2 for some a, b in Z. Now this is the norm of quaternion q where q=a1+bi+0j+0k since det (q)=a2+b2 but q is not one of q1,q2,...,qk since norm (q) is not any one of norm (q1), norm (q2), ..., norm (qk). Further q must be a quaternion prime since norm (q) is prime in Z. Therefore we have reached a contradiction. It follows that there must be infinitely many quaternion primes.