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.
