A review of the arithmetic of integers, division with remainder, and divisibility. Visual approaches to counting and summation. Many students will be comfortable skipping this chapter, but it's a nice warm-up.
The foundations of number theory, beginning with the Euclidean algorithm.
A thorough treatment of the Euclidean algorithm and the solution of linear Diophantine equations. Properties of the GCD and LCM.
The existence and uniqueness of prime factorization for positive integers. An exposition of the prime number theorem, Riemann hypothesis, conjectures and recent advances in the study of prime gaps. An introduction to multiplicative functions. Mersenne primes and perfect numbers.
A brief introduction to constructible numbers, followed by a thorough treatment of rational numbers. Rationality testing via prime decomposition and the rational root theorem. Mediant fractions and Ford circles. Diophantine approximation, including Dirichlet's theorem.
An introduction to complex numbers, followed by the arithmetic of the Gaussian integers and the Eisenstein integers. Includes uniqueness of prime decomposition, and relations between ordinary primes and Gaussian/Eisenstein primes (e.g., split, inert, and ramified primes). This chapter can be tackled simultaneously with Chapter 2, for advanced students.
A treatment of modular arithmetic through quadratic reciprocity, including applications to cryptography along the way.
An introduction to modular arithmetic. Divisibility by 3 and 9 tests. Multiplicative inverses, modulo m, and the solution of linear congruences. Polynomials mod p, including uniqueness of factorization.
The effect of repeated addition, or repeated multiplication, modulo m. Repeated multiplication by a, modulo m, introduces Fermat's little theorem and Euler's generalization (via the totient). Computation of the totient.
Connections between modular arithmetic with different moduli. The Chinese remainder theorem connects systems of congruences (modulo coprime moduli) to single congruences. A lifting technique connects certain congruences modulo a prime power to congruences modulo a higher power of the same prime. Lifting is introduced for linear congruences and for square roots modulo prime powers.
An introduction to quadratic residues and the Legendre symbol. A complete treatment of the sign of a permutation leads to Zolotarev's proof of quadratic reciprocity. (The sign of the "multiplication-by-a modulo p" permutation equals the Legendre symbol). Along the way, a proof of Fermat's Christmas Theorem (primes congruent to 1 mod 4 can be expressed as the sum of two squares).
A thorough treatment of integer-valued binary quadratic forms, using Conway's topograph to give visual proofs where possible.
The arithmetic of integer lattices, including primitive vectors and integer bases. Interpretations of the determinant. Primitive vectors are placed on a diagram called the "topograph" by J.H. Conway. An introduction to binary quadratic forms, and how they may be plotted on the topograph. The discriminant of a binary quadratic form, equivalence of forms, and isometries of a form.
Definite binary quadratic forms, including the solution of quadratic Diophantine equations. Reduction theory, expressed via Conway's "wells". Computation of class numbers for negative discriminants. Bounding the minimum nonzero value of a definite form.
Indefinite forms, including the solution of equations like Pell's equation. Reduction theory, expressed via Conway's "river". Infinitude of solutions to Pell's equation. Minimum value bound, computation of class numbers, and isometry groups. A brief introduction to the Markoff spectrum.