1.2 Course overview

One thing that you will surely learn from this course is that university mathematics has a rather different feel from that encountered at school. Here the emphasis is far more on proving general theorems than on performing calculations. One reason for this is that a result which can be applied to many different cases is clearly more powerful than a calculation which deals only with a single specific case. Another reason is that a general proof gives a much deeper insight into the problem under consideration than any number of isolated examples ever can.

Thus, we shall begin with a close look at the language and structure of mathematical proofs in general, highlighting the use of logic to express mathematical arguments in a concise and rigorous manner.

We shall then apply these ideas to the study of number theory which is the mathematical discipline concerned with properties of the whole numbers (or integers, as they are properly called) such as $3$, $-21$ or $769$.

It is easy to come up with questions about such numbers which are easy to ask, but look very difficult to answer; for instance,

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  what is the largest integer that exactly divides both $3\>470\>869\>254$ and $4\>179\>883\>216$?
  

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  can you find integers $r$ and $s$ such that $405r+262s=7\>938\>417$?
  

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  what is the smallest positive integer that can be exactly divided by both $3\>891\>762$ and $7\>528\>966$?
  

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  how many positive integers exactly divide $36\>000$?
  

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  what is the remainder on dividing $7^{42}$ by $13$?
  

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  what is the smallest positive integer that leaves remainder $79$ when divided by $101$, and remainder $246$ when divided by $324$?
  

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  is $815\>608\>943\>257$ a perfect square?
  

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  can $734\>098\>135\>903$ be written as a sum of two perfect squares?

In most cases, one could in principle answer the question by working through a very large number of possibilities, but this is hardly practical, and certainly not illuminating. We shall instead take an abstract approach by studying fundamental results in number theory, including Bézout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorizations, and we shall introduce a simplified form of integer arithmetic called modular arithmetic. Among other things, these results will enable us to tackle the questions listed above (and many other similar questions) in a systematic way.

However, it would be wrong to see this module as merely concerned with methods which enable us to solve ‘‘hard sums’’. What is of most importance to us is not simply the ability to perform impressive calculations, but the development of the underlying theory and the insight that it gives us into how numbers behave. This, in particular, will enable us to apply our methods and generalize our results to other areas of mathematics. We shall see two such examples towards the end of the course. First, the idea underlying the modular arithmetic mentioned above naturally leads to the abstract notion of an equivalence relation, a concept which has applications in many areas of mathematics (although we shall only be able to cover a couple of these). Second, we shall use our results in number theory to give meaning to the notion of a highest common factor of a pair of polynomials.