Index of Theorems – brain

Elementary Number Theory

Theorem	Location
Division Algorithm	2.4
Euclidean Algorithm	2.7
Extended Euclidean Algorithm	2.8
Bézout Identity	2.9
Euclid’s Lemma	3.1
Fundamental Theorem of Arithmetic	3.1
Infinitude of Primes	3.4
Chinese Remainder Theorem	4.7
Euler’s Theorem	5.3
Möbius Inversion Formula	5.7

Diophantine Equations

Theorem	Location
Solvability of Linear Diophantine Equations	6.1
Classification of Primitive Pythagorean Triples	6.2
Pell Equation Structure Theorem	6.3
Fermat Two-Squares Theorem	6.4
Rational Point Criterion for Conics	6.7

Quadratic Residues

Theorem	Location
Euler’s Criterion	7.4
Multiplicativity of the Legendre Symbol	7.2
Quadratic Reciprocity	7.5
Supplementary Laws of Quadratic Reciprocity	7.5
Properties of the Jacobi Symbol	7.3

Continued Fractions

Theorem	Location
Existence of Continued Fraction Expansions	8.2
Best Approximation Property of Convergents	8.5
Periodicity of Quadratic Irrational Continued Fractions	8.6
Pell Equation via Continued Fractions	8.6

Algebraic Number Theory

Theorem	Location
Existence of Algebraic Integer Rings	9.4
Multiplicativity of Norms	9.5
Ideal Factorization in Dedekind Domains	10.4
Finiteness of the Class Group	10.6
Minkowski Bound	10.8

Analytic Number Theory

Theorem	Location
Euler Product Formula	11.3
Abel Summation Formula	11.5
Chebyshev Bounds	12.2
Prime Number Theorem	12.3
Dirichlet’s Theorem on Arithmetic Progressions	14.4
Orthogonality of Dirichlet Characters	14.2

Zeta and $L$ -Functions

Theorem	Location
Analytic Continuation of $\zeta(s)$	13.3
Functional Equation of $\zeta(s)$	13.4
Explicit Formula for Prime Counting	13.7
Nonvanishing of $L(1,\chi)$	14.5
Generalized Riemann Hypothesis Statement	14.6

Additive Number Theory and Sieves

Theorem	Location
Cauchy-Davenport Theorem	15.1
Waring’s Problem	15.3
Hardy-Littlewood Circle Method Framework	15.4
Brun’s Theorem	16.1
Selberg Sieve Inequality	16.2
Large Sieve Inequality	16.3
Chen’s Theorem	16.5

Local and Global Fields

Theorem	Location
Hensel’s Lemma	18.4
Product Formula	18.5
Local-Global Principle for Quadratic Forms	18.5
Structure of Local Fields	18.2
Adele Decomposition Framework	18.6

Modular Forms and Elliptic Curves

Theorem	Location
Modular Transformation Law	20.3
Dimension Formula for Modular Forms	20.5
Hecke Eigenform Decomposition	20.6
Group Law on Elliptic Curves	21.2
Mordell’s Theorem	21.5
Hasse Bound	21.4
Birch and Swinnerton-Dyer Conjecture Statement	21.7

Arithmetic Geometry

Theorem	Location
Hilbert Nullstellensatz	22.1
Basic Properties of Projective Varieties	22.1
Scheme Gluing Principle	22.2
Properness and Finiteness Principles	22.3
Weil Conjectures	22.7

Automorphic Forms and Langlands

Theorem	Location
Strong Approximation Framework	23.3
Automorphic Representation Decomposition	23.2
Trace Formula Principle	23.6
Langlands Reciprocity Principle	23.4
Functoriality Conjecture	23.5

Computational Number Theory

Theorem or Algorithm	Location
Fast Modular Exponentiation	24.1
Miller-Rabin Primality Test	24.2
AKS Primality Test	24.2
Pollard Rho Factorization	24.3
Number Field Sieve	24.3
LLL Lattice Reduction	24.4
Schoof Point-Counting Algorithm	24.6

Cryptographic Number Theory

Theorem or Construction	Location
RSA Correctness	25.1
Diffie-Hellman Key Exchange	25.2
Elliptic Curve Discrete Logarithm Framework	25.3
Weil and Tate Pairings	25.4
Lattice-Based Cryptographic Hardness Assumptions	25.5
Zero-Knowledge Proof Completeness and Soundness	25.7

Open Problems and Conjectures

Problem or Conjecture	Location
Fermat’s Last Theorem	27.1
Catalan’s Conjecture	27.2
$abc$ Conjecture	27.3
Twin Prime Conjecture	27.4
Goldbach Conjecture	27.5
Riemann Hypothesis	27.6
Langlands Conjectures	27.7