Number Theory

This volume contains notes on number theory — the study of integers, particularly the natural numbers. What makes the integers so special? Well, perhaps the German mathematician Hermann Minkowski put it best:¹

Integral numbers are the fountainhead of all mathematics.

Reflecting on the history of mathematics reveals why Minkowski might have taken this position. The integers — barring debate on discovery or invention — came first.

It's not too difficult to see why. The natural numbers allowed early humans to enumerate: 1 apple, 2 hogs, 3 swords, 4 houses, and so on. As time progressed, human societies grew larger, and the spaces they occupied widened. With more resources and people, these ancient societies were confronted with increasingly complex computations — surveying land to resolve conflicts, tracking time, dividing common resources, calculating tributes, etc.

The methods for performing these computations — with just the natural numbers — helped provide some of the foundations of what we now know as algebra. Occassionally, the mathematicians of antiquity encountered peculiar questions:

To solve these questions, mathematicians created a new set of numbers, ${\uint.}$ Not too long after, another peculiar question appeared:

Once again, the mathematicians created another set of numbers: ${\rat.}$ Then the mathematicians encountered another strange question:

Once more, another set emerged — ${\reals.}$ Then the mathematicians encountered perhaps the strangest question they'd seen yet:

It took quite some time for mathematicians to come to accept the set of complex numbers, ${\com.}$ These new sets have generated their own fields — the reals are studied in real analysis, and the complex numbers in complex analysis. Number theory, however, stays at the root of it all: the natural numbers. As it turns out, there are numerous questions that remain unanswered. This volume presents just a few of these questions.