Volume 2

Five is a sum of two integer squares. Seven is not. What decides the difference?

Every integer square leaves remainder 0 or 1 on division by 4. That rules out 7 as a sum of two squares. But why can every prime leaving remainder 1 be represented? An obstruction and a construction need different arguments.

The Integers: Primes, Patterns and Proofs is Volume 2 of Order and Chance: The Hidden Architecture of Mathematics.

Its four parts and eleven chapters build number theory from Euclid's algorithm, unique prime factorisation, congruences and arithmetic functions.

Primitive roots describe cycles of remainders. Quadratic reciprocity compares square residues at different primes. Infinite descent rules out integer solutions, while continued fractions construct solutions of Pell's equation. Gaussian and Eisenstein integers supply new factorisations. Dirichlet characters, Euler products and elementary sieves connect prime factors with sums and counts, with the limits of those methods stated explicitly.

Worked examples and diagrams accompany 144 graded exercises, each with a hint and complete solution. The first-reading route identifies the foundations; optional advanced problems combine tools already taught on their reading route.

An Arithmetic Field Guide gathers definitions and helps you choose a method. The prerequisite is Volume 1, First Principles. No abstract algebra course is assumed. New arithmetic concepts are defined before they are used in proofs.

The next volume, Symmetry: Groups, Rings and Fields, studies the structures behind these examples. Deeper results about prime distribution, general ideal theory and p-adic numbers belong to later volumes.