Number Theory
Why are the simplest questions the hardest?
Every question in this subject can be explained to a child and several have taken centuries. That gap between the statement and the proof is the reason number theory is where most mathematicians learn what difficulty actually feels like.
Divisibility
How much is hiding in long division?
The oldest algorithm still in use
Euclid's algorithm finds a greatest common divisor without factorising anything, and the reason it works is more interesting than the algorithm.
Why factorisation is unique
That every number factors into primes one way only is so familiar it looks like a definition. It is a theorem, and it fails in nearby systems.
Congruences
What if you only care about remainders?
Arithmetic on a clock
Discarding everything but the remainder ought to lose information, and mostly it loses exactly the information that was in the way.
Fermat, Euler, and a theorem you can use
Two results about remainders that between them make modern cryptography possible — proved here, then applied.
Distribution of primes
Is there a pattern, or only the absence of one?
Primes thin out, but never stop
Euclid's proof of infinitude is four lines. The question of how the primes are spread out is still open in places, and the film is honest about where.
Counting primes up to a million
The prime counting function looks random up close and smooth from far away. The document proves what can be proved and says clearly what cannot.
Quadratic reciprocity
Why should two primes know anything about each other?
Number Theory, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume IX ↓