Number theory

The arithmetic of the integers: divisibility, primes, congruences, and the equations whose solutions are required to be whole numbers. Until 2026-10-04 this spoke covered it only inside abstract-algebra, where lal-algebra-1 and judson-abstract-algebra treat it as ring theory. baker-algebra-number-theory is the first source here that teaches it as a subject in its own right, and it does so before any ring appears.

The chain in Baker's chapter 1

1. Well-ordering, induction and the maximal principle are proved equivalent (Theorem 1.1), and the long-division property follows from the maximal principle.
2. The Euclidean algorithm gives the gcd and, by back-substitution or a tabular method, integers u, v with au + bv = gcd(a, b). Everything later leans on this one identity.
3. Congruences and the units of ℤ/n: t is invertible exactly when gcd(t, n) = 1. The Chinese remainder theorem is stated for two coprime moduli.
4. Primes: Euclid's lemma, unique factorization, infinitely many primes, irrationality of √p.
5. Congruences modulo a prime: a polynomial of degree d has at most d roots, then Fermat's little theorem, a "binomial theorem" for (u + v)^p mod p (called the Idiot's Binomial Theorem in the notes), a primitive root, and Wilson's theorem.
6. Continued fractions, finite then infinite: convergents, uniqueness for irrationals, and the periodic expansion of √n for non-square n, with the period listed for many n (period 11 for √97).
7. Diophantine equations: ax + by = c solved by the Euclidean algorithm.
8. Pell's equation x² − dy² = 1: every positive solution is a convergent of the continued fraction of √d, and all solutions are powers of the fundamental one. The notes work x² − 2y² = 1 (solutions (3, 2), (17, 12), (99, 70)) and x² − 3y² = 1 (solutions (2, 1), (7, 4), (26, 15)).

The two ends are the point: the course starts from the principle that the integers can be well-ordered and ends with an equation, Pell's, whose complete solution needs the continued-fraction machinery of the middle.

Arithmetic functions (chapter 3)

A function on the positive integers, with Dirichlet convolution as a product. The constant function 1 has the Möbius function µ as its inverse under convolution, and Möbius inversion follows: if g = f * 1 then f = g * µ. The notes use it to recover Euler's φ from n = Σ φ(d) over d | n, and the problem set asks for identities among σ, τ and the power functions. Multiplicativity is the working property: the convolution of two multiplicative functions is multiplicative.

Where it meets the rest of the spoke

• group-theory: the units of ℤ/p form a cyclic group, and Baker proves this by the group-theory theorem above. Lagrange's theorem then gives the order of any element dividing the group order.
• abstract-algebra: the same material arrives there as ring theory. The two orderings (arithmetic first, rings first) are both now held, which lets the corpus compare them.
• cardinality-and-countability: the last chapter of the same notes.

Open

No source here treats quadratic reciprocity, quadratic residues or anything analytic; lal-algebra-1 has the first two in "Number Theory 2." Primality testing and factoring, the computational side, are absent, and so is the cryptographic application, which sits in ../cryptography-wiki's territory.