# Index of Definitions

## Foundations

| Definition | Location |
|---|---|
| Natural Number | 1.1 |
| Integer | 1.2 |
| Absolute Value | 1.5 |
| Mathematical Induction | 1.6 |
| Strong Induction | 1.7 |
| Recursive Definition | 1.8 |

## Divisibility

| Definition | Location |
|---|---|
| Divisibility | 2.1 |
| Prime Number | 2.2 |
| Composite Number | 2.3 |
| Greatest Common Divisor | 2.5 |
| Least Common Multiple | 2.6 |
| Coprime Integers | 2.10 |

## Prime Factorization

| Definition | Location |
|---|---|
| Prime Factorization | 3.1 |
| Canonical Prime Decomposition | 3.2 |
| Prime Counting Function | 12.1 |
| Prime Gap | 12.7 |

## Congruences

| Definition | Location |
|---|---|
| Congruence Modulo $n$ | 4.1 |
| Residue Class | 4.2 |
| Modular Inverse | 4.5 |
| System of Congruences | 4.6 |

## Arithmetic Functions

| Definition | Location |
|---|---|
| Arithmetic Function | 5.1 |
| Divisor Function | 5.1 |
| Möbius Function | 5.2 |
| Euler Totient Function | 5.3 |
| Liouville Function | 5.4 |
| Multiplicative Function | 5.5 |
| Dirichlet Convolution | 5.6 |

## Diophantine Equations

| Definition | Location |
|---|---|
| Diophantine Equation | 6.1 |
| Pythagorean Triple | 6.2 |
| Pell Equation | 6.3 |
| Rational Point | 6.7 |
| Integral Point | 6.7 |

## Quadratic Residues

| Definition | Location |
|---|---|
| Quadratic Residue | 7.1 |
| Quadratic Nonresidue | 7.1 |
| Legendre Symbol | 7.2 |
| Jacobi Symbol | 7.3 |
| Gauss Sum | 7.6 |

## Continued Fractions

| Definition | Location |
|---|---|
| Continued Fraction | 8.1 |
| Convergent | 8.5 |
| Quadratic Irrational | 8.6 |
| Diophantine Approximation | 8.7 |

## Algebraic Number Theory

| Definition | Location |
|---|---|
| Algebraic Integer | 9.1 |
| Minimal Polynomial | 9.2 |
| Number Field | 9.3 |
| Ring of Integers | 9.4 |
| Norm | 9.5 |
| Trace | 9.5 |
| Unit | 9.6 |
| Ideal | 9.8 |
| Principal Ideal | 10.1 |
| Prime Ideal | 10.2 |
| Maximal Ideal | 10.2 |
| Fractional Ideal | 10.3 |
| Dedekind Domain | 10.5 |
| Class Group | 10.6 |
| Class Number | 10.7 |

## Analytic Number Theory

| Definition | Location |
|---|---|
| Infinite Series | 11.1 |
| Infinite Product | 11.2 |
| Euler Product | 11.3 |
| Abel Summation | 11.5 |
| Chebyshev Functions | 12.2 |
| Error Term | 12.5 |
| Short Interval | 12.6 |

## Zeta and $L$-Functions

| Definition | Location |
|---|---|
| Riemann Zeta Function | 13.1 |
| Nontrivial Zero | 13.5 |
| Critical Strip | 13.5 |
| Critical Line | 13.6 |
| Dirichlet Character | 14.1 |
| Orthogonality Relation | 14.2 |
| Dirichlet $L$-Function | 14.3 |

## Additive and Sieve Methods

| Definition | Location |
|---|---|
| Sumset | 15.1 |
| Goldbach Problem | 15.2 |
| Waring Problem | 15.3 |
| Circle Method | 15.4 |
| Brun Sieve | 16.1 |
| Selberg Sieve | 16.2 |
| Large Sieve | 16.3 |

## Algebraic Extensions and Local Fields

| Definition | Location |
|---|---|
| Field Extension | 17.1 |
| Splitting Field | 17.2 |
| Galois Group | 17.3 |
| Finite Field | 17.4 |
| Cyclotomic Field | 17.5 |
| Ramification | 17.6 |
| Absolute Value | 18.1 |
| $p$-Adic Number | 18.2 |
| Completion | 18.3 |
| Local Field | 18.5 |
| Adele | 18.6 |
| Idele | 18.6 |

## Class Field Theory

| Definition | Location |
|---|---|
| Abelian Extension | 19.1 |
| Reciprocity Map | 19.2 |
| Hilbert Class Field | 19.3 |
| Local Class Field Theory | 19.4 |
| Global Class Field Theory | 19.5 |

## Modular Forms

| Definition | Location |
|---|---|
| Modular Group | 20.1 |
| Modular Function | 20.2 |
| Modular Form | 20.3 |
| Eisenstein Series | 20.4 |
| Cusp Form | 20.5 |
| Hecke Operator | 20.6 |
| Modular Curve | 20.7 |

## Elliptic Curves

| Definition | Location |
|---|---|
| Plane Cubic Curve | 21.1 |
| Elliptic Curve | 21.1 |
| Point at Infinity | 21.2 |
| Group Law | 21.2 |
| Rational Point | 21.3 |
| Mordell-Weil Group | 21.5 |

## Arithmetic Geometry

| Definition | Location |
|---|---|
| Affine Variety | 22.1 |
| Projective Variety | 22.1 |
| Scheme | 22.2 |
| Morphism | 22.3 |
| Fiber | 22.3 |
| Arithmetic Surface | 22.5 |
| Étale Cohomology | 22.6 |

## Automorphic Forms and Representations

| Definition | Location |
|---|---|
| Representation | 23.1 |
| Automorphic Representation | 23.2 |
| Adelic Method | 23.3 |
| Langlands Program | 23.4 |
| Functoriality | 23.5 |
| Trace Formula | 23.6 |

## Computational and Cryptographic Number Theory

| Definition | Location |
|---|---|
| Primality Test | 24.2 |
| Integer Factorization Algorithm | 24.3 |
| Lattice Reduction | 24.4 |
| RSA Cryptosystem | 25.1 |
| Diffie-Hellman Key Exchange | 25.2 |
| Elliptic Curve Cryptography | 25.3 |
| Pairing-Based Cryptography | 25.4 |
| Lattice Cryptography | 25.5 |
| Post-Quantum Cryptography | 25.6 |
| Zero-Knowledge Proof | 25.7 |