Number Theory
The integers, viewed up close. Primes, divisibility, congruences, Diophantine equations — and the strange depth that the simplest objects in mathematics can have. The Riemann Hypothesis is a question about the natural numbers.
Modular arithmetic — the workhorse of elementary number theory — lives in Number Systems › Modular Arithmetic.
Elementary Number Theory
Divisibility, primes, GCD, the Euclidean algorithm (and its Bézout-producing extension), the Fundamental Theorem of Arithmetic, and Euclid's infinitude proof.
Arithmetic Functions & Primes
Multiplicative functions (τ, σ, φ, μ), Möbius inversion, the Riemann zeta function and its Euler product, the Prime Number Theorem, and the Riemann Hypothesis.
Quadratic Reciprocity
The gem of elementary number theory. Quadratic residues, the Legendre symbol, Euler's criterion, and Gauss's law of reciprocity.
Diophantine Equations
Equations seeking integer solutions. Linear, Pythagorean triples, Pell's equation, Fermat's Last Theorem, and Hilbert's tenth problem.