Now it’s time to introduce maybe the most important concept in the whole course. It’s one you are almost certainly already pretty familiar with. That is the concept of prime numbers.
Although we’ll take a somewhat traditional route to introduce them, consider what precedes this chapter. We attacked linear congruences as far as we could via the concept of ‘relatively prime’/‘coprime’. But the thought should be gnawing at us of whether there is something deeper than simply not sharing factors other than one; what are the factors that are (or are not) shared in the first place? As mathematicians, we always want to ask whether there is a simpler notion available, or one that explains more.
We will see the fruit of this for linear congruences in Section 6.5
, using the most powerful tool in our arsenal, Theorem 6.3.2
. But once we have unleashed the power of primes, we will see and use them everywhere, such as in Chapters 22
. Examining them more closely will lead to us some of the deepest mathematics of the book in Chapters 21
So let’s get started!