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Modern public-key cryptography relies heavily on two computational assumptions:

The Langlands program is one of the most ambitious and influential theories in modern mathematics.

Exponential sums are among the central tools of analytic number theory.

In ordinary analysis, the absolute value

Ordinary integers satisfy several remarkable properties simultaneously:

Study empirical properties of prime numbers through computation.

Lattice cryptography is a family of cryptographic systems based on the presumed hardness of computational problems on high-dimensional lattices.

Many problems in additive number theory ask whether an integer can be represented in the form

Classical modular form theory begins with analytic functions satisfying symmetry conditions.

Let $R$ be a commutative ring. An ideal $I\subseteq R$ is called principal if there exists an element $\alpha\in R$ such that

Waring's problem asks whether every sufficiently large positive integer can be written as a sum of a bounded number of fixed powers.

Pairing-based cryptography uses special maps defined on elliptic curve groups. A pairing is a function

Modular forms are functions on the upper half-plane satisfying symmetry conditions under the modular group

The discriminant is one of the most important invariants of a number field. It measures how the arithmetic of the field differs from ordinary rational arithmetic.

Goldbach-type problems ask whether integers can be represented as sums of primes. They are among the oldest and most famous problems in additive number theory.

Elliptic curve cryptography is a public-key cryptographic framework based on the arithmetic of elliptic curves over finite fields.

For centuries, elliptic curves and modular forms were studied as separate objects.

Let $K$ be a number field and let

In ordinary integers, every ideal is generated by a single element:

Secure communication requires two parties to share secret information. In classical symmetric cryptography, both parties must already possess the same secret key before...

Additive number theory studies arithmetic structure through addition of integers and subsets of integers.

An elliptic curve is simultaneously:

1. Prove that the sum of two even integers is even.

The classical Riemann Hypothesis concerns the zeros of the Riemann zeta function

In ordinary integers, every number factors uniquely into primes. In many rings of algebraic integers, this property fails.

One of the central properties of the ordinary integers is unique factorization.

Let $K$ be a number field of degree

Classical cryptography uses a shared secret key. Both sender and receiver must know the same secret information in advance.

The modular group acts on the upper half-plane by fractional linear transformations:

A central theme in analytic number theory is determining when an $L$-function is nonzero at a particular point.

Modern number theory relies heavily on computation. Two broad computational paradigms dominate the subject:

An arithmetic progression is a sequence of the form

Modular forms already possess symmetry under the modular group. Yet a deeper arithmetic structure emerges through another family of operators: the Hecke operators.

In ordinary arithmetic, the integers

The Riemann zeta function

Elliptic curves occupy a central position in modern number theory, arithmetic geometry, and cryptography.

Modular forms satisfy strong symmetry conditions under the modular group. Among them, cusp forms form the deepest and most arithmetic subclass.

Modular forms are highly structured analytic functions with deep arithmetic properties. Although their definitions involve complex analysis and group actions, modular forms...

Among all modular forms, Eisenstein series are the most explicit and computationally accessible.

Dirichlet characters behave analogously to exponential functions in Fourier analysis. Just as complex exponentials separate frequencies, characters separate residue classes...

A number field is a finite extension of the rational numbers. Concretely, it is a field $K$ satisfying

Modular forms are among the central objects of modern number theory.

A lattice is a discrete additive subgroup of Euclidean space. More concretely, let

| Period | Development |

The Riemann zeta function studies prime numbers globally, without distinguishing congruence classes. However, many arithmetic questions concern primes satisfying conditions such as

The modular group acts on the upper half-plane by fractional linear transformations:

An algebraic number is a complex number that satisfies some nonzero polynomial equation with rational coefficients. Thus $\alpha\in\mathbb{C}$ is algebraic if there exists a...

| Definition | Location |

The Riemann zeta function was introduced through the series

Integer factorization asks for the prime decomposition of a positive integer. Given

Euler products are one of the central ideas of analytic number theory. They express infinite sums over integers as infinite products over primes.

An arithmetic function $fn$ can be encoded into an infinite series of the form

Modular forms begin with the action of certain matrix groups on the complex upper half-plane.

The ordinary integers

| Theorem | Location |

A prime number is an integer greater than $1$ whose only positive divisors are

One of the central goals of algebraic number theory is to classify field extensions of a number field

One of the deepest ideas in analytic number theory is that the zeros of the zeta function determine the distribution of prime numbers.

Diophantine approximation studies how closely real numbers can be approximated by rational numbers.

Arithmetic functions often fluctuate strongly from one integer to the next.

Many arithmetic functions are defined through sums over divisors. For example,

Recall that a Pell equation has the form

The Riemann zeta function has nontrivial zeros inside the critical strip

Modern computational number theory depends fundamentally on efficient arithmetic with large integers.

Global class field theory studies finite abelian extensions of number fields such as

Arithmetic functions can be added and multiplied pointwise, but number theory has another product that is better adapted to divisibility.

The convergents of a continued fraction are the rational numbers obtained by truncating the expansion at finite stages.

The zeros of the Riemann zeta function are the complex numbers $s$ satisfying

| Symbol | Meaning |

One of the central ideas of modern analysis is that functions may be decomposed spectrally into elementary pieces.

One of the central discoveries of algebraic number theory is that unique factorization may fail in rings of algebraic integers.

The defining series of the zeta function,

Many important numbers are irrational:

An arithmetic function is a function defined on the positive integers. Such a function

The Liouville function is an arithmetic function denoted by

The defining series of the Riemann zeta function is

One of the oldest themes in number theory is reciprocity: the phenomenon that solvability conditions for one prime are controlled by arithmetic involving another prime.

Finite continued fractions correspond exactly to rational numbers. When the Euclidean algorithm never terminates, the continued fraction becomes infinite.

Functoriality is the unifying mechanism of the Langlands program. It predicts systematic relationships between automorphic representations attached to different algebraic groups.

Euler's totient function is an arithmetic function denoted by

The defining series of the Riemann zeta function is

A central goal of algebraic number theory is to understand field extensions of a given base field, especially extensions of the rational numbers

A finite continued fraction is an expression of the form

A group $G$ is abelian if

One of the central objects of analytic number theory is the Riemann zeta function. It connects infinite series, prime numbers, complex analysis, and arithmetic structure into...

The Euclidean algorithm is one of the oldest and most important algorithms in mathematics. It computes the greatest common divisor of two integers using repeated division.

The rational numbers may be studied through their completions:

The Möbius function is an arithmetic function denoted by

The Langlands program is a broad collection of conjectures connecting number theory, representation theory, harmonic analysis, and algebraic geometry. Its central idea is that...

A pair of primes

Quadratic residue theory is not only a theoretical subject. It also plays a major role in computational number theory, cryptography, primality testing, and algorithm design.

A central problem in number theory is determining whether an equation possesses rational or integral solutions.

Number theory is one of the oldest parts of mathematics, but modern number theory is not a single ancient subject carried forward unchanged. It is a layered discipline....

Divisor functions measure the positive divisors of an integer. They are among the first examples of arithmetic functions, because their values depend directly on the prime...

Modular arithmetic is not only a theoretical language for divisibility. It is also one of the main tools of computation with integers.

Let

Number theory studies arithmetic simultaneously at two levels:

Quadratic reciprocity describes when one prime is a square modulo another prime. A natural question is whether similar laws exist for higher powers.

One of the central ideas of number theory is that congruences modulo powers of a prime often approximate genuine arithmetic solutions.

Modular arithmetic often requires computing powers such as