Number Theory

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

1. Part 1 · Divisibility and primes

   4 pages

   Number theory here starts from the integers as an ordered commutative ring with a division algorithm, so gcds, Bezout's identity and Euclid's lemma lead to unique prime factorisation. The same repeated quotient extraction becomes continued fractions: convergents satisfy determinant identities and sharp approximation bounds, Legendre's criterion characterises best approximants, and eventual periodicity detects quadratic irrationals. For $\sqrt{D}$ that periodicity becomes arithmetic in $\mathbb{Z}[\sqrt{D}]$: the norm is multiplicative, Pell solutions form a unit group generated by the fundamental solution, parity of the continued-fraction period decides the negative Pell equation, and every nonzero generalized Pell equation breaks into finitely many orbits, making solubility decidable by a finite search.

   • level 0Divisibility, Greatest Common Divisors and Bézout's Identity26 results

     This page opens the number theory track, and it is built entirely on ℤ as the library constructed it: the ordered commutative ring of The integers form a commutative ring and…

     6 definitions, 12 lemmas, 4 theorems, 4 corollariesExamples & counterexamples
   • level 1Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic18 results

     Divisibility in ℤ, Bézout’s identity, the gcd characterisation, and the coprime-divides-product lemma provide the arithmetic background.

     2 definitions, 6 lemmas, 6 theorems, 4 corollariesExamples & counterexamples
   • level 1Regular Continued Fractions and Diophantine Approximation19 results

     Regular continued fractions turn Euclidean division into an approximation machine.

     5 definitions, 6 lemmas, 7 theorems, 1 corollaryExamples & counterexamples
   • level 2Pell Equations and Generalized Pell Orbits17 results

     Pell's equation is where the continued-fraction machinery of √D stops being a theory of approximation and becomes a machine for producing integer solutions.

     5 definitions, 3 lemmas, 1 proposition, 5 theorems, 3 corollariesExamples & counterexamples

2. Part 2 · Congruences and unit groups

   11 pages · after Part 1

   Congruences, Chinese remainder theorem, Euler's totient and primitive roots give quotient-ring arithmetic; quadratic forms and arithmetic functions organize multiplicative information. For finite $K\mathrm{/}\mathbb{Q}$, trace, norm, embeddings and discriminants give the Dedekind property and unique prime-ideal factorization; the different detects ramification (exponent $e-1$ tame), and completions supply ${\mathbb{Q}}_p$ and ${\mathbb{Z}}_p$. The decomposition page identifies the decomposition group as prime stabilizer and inertia as residue-action kernel, reading ramification, residue degree, splitting and Frobenius cycle lengths from exact sequences, towers, fixed fields and cosets. Minkowski embedding and convex bodies turn ${\mathcal{O}}_K$ and ideals into full lattices, bound discriminants and force the class group finite; unit logarithms form a full lattice whose rank and regulator give Dirichlet's theorem and $S$-units.

   • level 2Congruences, the Integers Modulo n and the Chinese Remainder Theorem23 results

     Congruence turns divisibility into an equivalence relation and hence into the quotient ℤ/n, where addition and multiplication are independent of representatives.

     4 definitions, 7 lemmas, 10 theorems, 2 corollariesExamples & counterexamples
   • level 3Absolute Values Completions and P Adic Numbers22 results

     This page classifies the rank-one absolute values on ℚ, records the place language and the rational product formula, and then builds ℚ p as a completion before comparing it with the integral compatible-residue model.

     5 definitions, 2 lemmas, 11 theorems, 4 corollariesExamples & counterexamples
   • level 3Arithmetic Functions and Dirichlet Convolution19 results

     Arithmetic functions package number-theoretic data into maps on the positive integers.

     10 definitions, 2 propositions, 5 theorems, 2 corollariesExamples & counterexamples
   • level 3Number Fields Rings of Integers and Discriminants20 results

     For a finite extension K/ℚ, this page separates its maximal order O K from arbitrary full-rank orders.

     6 definitions, 1 lemma, 7 theorems, 6 corollariesExamples & counterexamples
   • level 3Positive Definite Binary Quadratic Forms and Reduction23 results

     Integral binary quadratic forms are the ternary coefficient data (a,b,c) behind ax²+bxy+cy², and this page studies how much of that data survives unimodular change of variables.

     9 definitions, 5 lemmas, 4 propositions, 3 theorems, 2 corollariesExamples & counterexamples
   • level 3Primitive Roots and Unit Groups Modulo N28 results

     The published unit group (ℤ/nℤ)^× collects the residue classes coprime to n, the unit criterion recognises them, and Euler's totient counts them.

     3 definitions, 7 lemmas, 3 propositions, 7 theorems, 8 corollariesExamples & counterexamples
   • level 4Dirichlet Series and Euler Products15 results

     This page fixes Dirichlet-series notation, proves the half-plane and abscissa geometry, and then turns multiplicativity into Euler products only in regions where absolute convergence genuinely licenses the regrouping.

     2 definitions, 9 theorems, 4 corollariesExamples & counterexamples
   • level 4Prime Ideal Decomposition Ramification and the Different24 results

     For number fields, integral ideal factorisation is handled through a finite quotient and finite local data.

     7 definitions, 2 lemmas, 11 theorems, 4 corollariesExamples & counterexamples
   • level 5Decomposition Inertia and Frobenius29 results

     Under the Axiom of Choice, completions and unique extension of nonarchimedean absolute values provide the local bridge.

     5 definitions, 6 lemmas, 14 theorems, 4 corollariesExamples & counterexamples
   • level 5Minkowski Theory and Number Field Class Groups24 results

     The unscaled Minkowski embedding turns O K and its nonzero fractional ideals into full Euclidean lattices of covolume 2^-r 2√|d K| Na in ℝ^r 1+2r 2.

     3 definitions, 9 lemmas, 8 theorems, 4 corollariesExamples & counterexamples
   • level 6Dirichlets Unit Theorem Regulators and S Units18 results

     The unit group of a number field is finite at rank zero and otherwise a finite torsion group times a free abelian group.

     4 definitions, 7 lemmas, 6 theorems, 1 corollaryExamples & counterexamples

3. Part 3 · Quadratic residues and reciprocity

   12 pages · after Part 2

   The Legendre and Jacobi symbols, Euler's criterion, Gauss's lemma, and reciprocity control quadratic residues and the two- and four-square problems. Average orders and Perron inversion lead to Chebyshev and Mertens estimates and the explicit formula, while Hilbert symbols give Hasse--Minkowski and Dirichlet characters give primitive $L$-function continuation and functional equations. Cyclotomic arithmetic gives the ring of integers $\mathbb{Z}[{\zeta }_n]$ and its discriminant, the ramification and Frobenius decomposition of rational primes, and reciprocity via Gauss sums. The zeta page uses the Hadamard product and a logarithmic-derivative inequality for a zero-free region near $1$; a balanced truncated explicit formula yields the stated prime-number-theorem error. Its damped-contour Tauberian route is independent, and the progression result keeps the modulus fixed.

   • level 4Quadratic Residues and the Legendre Symbol16 results

     For a prime modulus, the nonzero residue classes form a cyclic unit group of order p-1.

     2 definitions, 1 lemma, 3 propositions, 7 theorems, 3 corollariesExamples & counterexamples
   • level 5Quadratic Reciprocity and the Jacobi Symbol18 results

     Gauss's lemma turns a Legendre symbol into the parity of a finite lower-half count.

     1 definition, 3 lemmas, 3 propositions, 9 theorems, 2 corollariesExamples & counterexamples
   • level 5Sums of Two Squares18 results

     The first supplement to quadratic reciprocity determines exactly when -1 is a square modulo an odd prime.

     1 definition, 7 lemmas, 2 propositions, 4 theorems, 4 corollariesExamples & counterexamples
   • level 6Average Orders Divisor Sums and Representation Counts17 results

     This page fixes the summatory notion of average order and then carries the standard finite-sum arguments that make the first analytic-number-theory constants visible: the…

     3 definitions, 3 lemmas, 6 theorems, 5 corollariesExamples & counterexamples
   • level 6Cyclotomic Arithmetic and Reciprocity via Frobenius21 results

     This page develops the arithmetic of the full cyclotomic fields ℚ(ζ n) for reduced indices : an index n is reduced when it is odd or divisible by 4.

     2 definitions, 5 lemmas, 7 theorems, 7 corollariesExamples & counterexamples
   • level 6Lagrange Four Square Theorem16 results

     Division with remainder for a nonzero divisor, the divisibility relation with its linearity and transitivity, and congruence modulo an integer together with its compatibility…

     1 definition, 9 lemmas, 1 proposition, 2 theorems, 2 corollaries, 1 remarkExamples & counterexamples
   • level 7Chebyshev Bounds and Mertens Theorems18 results

     This page develops the classical finite arguments behind Chebyshev's density theorem, Bertrand's postulate, and the three Mertens theorems.

     4 definitions, 6 lemmas, 6 theorems, 2 corollariesExamples & counterexamples
   • level 8Dirichlet Characters L Functions and Primes in Progressions24 results

     Dirichlet characters are the Fourier characters of the finite group (ℤ/qℤ)^×, written as arithmetic functions by extending them by zero off the units.

     4 definitions, 8 lemmas, 11 theorems, 1 corollaryExamples & counterexamples
   • level 8Perron Inversion and the Explicit Formula14 results

     Symmetric Perron inversion gives half of a coefficient at a jump.

     3 definitions, 5 lemmas, 5 theorems, 1 corollaryExamples & counterexamples
   • level 9Classical Zero Free Region and the Prime Number Theorem20 results

     Under countable choice, the Hadamard product and the nonnegative trigonometric polynomial give the classical zero-free region 1-β≫1/log(|γ|+3).

     1 definition, 9 lemmas, 6 theorems, 4 corollariesExamples & counterexamples
   • level 9Hilbert Symbols and the Quadratic Local Global Principle17 results

     The rational local fields are the archimedean completion ℝ together with the p-adic fields ℚ p.

     2 definitions, 5 lemmas, 8 theorems, 2 corollariesExamples & counterexamples
   • level 9Primitive Dirichlet L Functions and Functional Equations16 results

     This page fixes e(x)=exp(2π i x) and f̂(ξ)=∫ ℝf(x)e(-xξ) dx. It develops primitive characters, their Gauss sums, and the parity-sensitive theta argument leading to the…

     5 definitions, 2 lemmas, 7 theorems, 2 corollariesExamples & counterexamples