# Factors of a prime squared minus one — a surprising result?

This is a past question from the Cambridge University Maths entrance interview. It is an interesting open-ended question with quite a surprising result. Consider the following value *n*:

Where *p* is a prime number greater than 3.

The question is: what can you say about the prime factors of *n*?

Think about it before looking at the answer.

# Tackling the problem

Since this is a question about the prime factors of *n*, a good place to start might be to factorise *n* itself. We can see that *n* is a difference of two squares:

This can be factorised into:

So we now have the slightly easier problem of finding the prime factors of *(p — 1)* and *(p + 1)*.

# What can we tell about the factors?

What do we know about the two factors? There are two obvious things we can say:

*(p — 1)*and*(p + 1)*differ by 2.*(p — 1)*,*p*, and*(p + 1)*are consecutive.

Not forgetting, of course, we have been told that *p* is prime and greater than 3.