1.1 Introduction
In this course we will study numbers, more precisely we will study integers.
The integer number system has the property that we can add and subtract numbers, and can also multiply them, but we cannot in general divide integers. With this in mind we will try to avoid writing divisions in number theory: if \(a\) and \(b\) are integers for which \(a\) is divisible by \(b\), and we want to consider the ratio \(x\) of the two numbers then instead of writing \(x=a/b\), we would write \(a=bx\). This avoids the possibility of writing down an expression like \(2/3\) which is not defined in the context of the integers.
Our aim in the course is to prove results about the integers. In school you may have seen a few proofs but throughout university, especially in the area of pure mathematics, you will find a much greater emphasis on proof. In this module we will give an introduction to logic and proof techniques.
Why is it important to prove results? Here are some examples of results that we can prove in number theory.
- \(p\) is a prime number if and only if \((p-1)!+1\) is divisible by \(p\).
- For any integers \(a,b,c\) the equation \[ax+by=c\] has integer solutions if and only if \(c\) is divisible by all common factors of \(a\) and \(b\).
- There are infinitely many integer solutions to the equation\[a^2+b^2=c^2\]where \(a,b,c\) have no common factors.
- For \(p\) a prime other than \(2\), the equation\[n^2+1=kp\] has integer solutions if and only if \(p-1\) is divisible by \(4\).
These are examples of Theorems from number theory. A theorem is a statement which we know to be true because it has been proved.
Results such as these are true and may be useful, but are not obvious. Proving them allows us to be sure that the result is always true and, once we know this, we can then use these results as facts to help answer other questions.
On the other hand we might spot a pattern and think that a result is always true, but without a proof we cannot be sure. As an example, consider the numbers \(2^p-1\) where \(p\) is a prime.
| \(p\) | \(2^p-1\) |
|---|---|
| 2 | 3 |
| 3 | 7 |
| 5 | 31 |
| 7 | 127 |
| 11 | 2047 |
The observation that 3, 7, 31 and 127 are themselves all prime numbers might lead us to think that if \(p\) is a prime then \(2^p\) must also be prime. It might even convince us that 2047 is also prime, after all there are no obvious factors: it is easy to see it is not divisible by 2, 3, 5, 7, 11, and computing a bit further it is also not divisible by 13, 17, 19. But 2047 is not prime (you don’t have to go much further to find a factor).
Consider this as a warning: Looking for patterns can be very valuable and instructive, but without a proof you cannot be sure if the pattern continues.