Glossary of number theory

Last updated July 25, 2024

This is a glossary of concepts and results in number theory, a field of mathematics. Concepts and results in arithmetic geometry and diophantine geometry can be found in Glossary of arithmetic and diophantine geometry.

A

abc: abc conjecture
adele: Adele ring
Algebraic number: Algebraic number
Algebraic number field: See number field.
Algebraic number theory: Algebraic number theory
Analytic number theory: Analytic number theory
Artin: The Artin conjecture says Artin's L function is entire (holomorphic on the entire complex plane).
automorphic form: An automorphic form is a certain holomorphic function.

B

Bézout's identity: Bézout's identity, also called Bézout's lemma, states that if $d$ is the greatest common divisor of two integers $a$ and $b$ , then there exists integers $x$ and $y$ such that $ax + by = d$ , and in fact the integers of the form $as + bt$ are exactly the multiples of $d$ .
Brocard: Brocard's problem

C

Chinese remainder theorem: Chinese remainder theorem
class field: The class field theory concerns abelian extensions of number fields.
class number: 1. The class number of a number field is the cardinality of the ideal class group of the field.; 2. The class number problem.
conductor: conductor (number theory)
Coprime: Two integers are coprime if the only positive integer that divides them both is 1.

D

Dedekind: Dedekind zeta function.
Diophantine equation: Diophantine equation
Dirichlet: 1. Dirichlet's theorem on arithmetic progressions; 2. Dirichlet character; 3. Dirichlet's unit theorem.
Disquisitiones: Disquisitiones Arithmeticae is a book by Carl Friedrich Gauss.
distribution: A distribution in number theory is a generalization/variant of a distribution in analysis.
divisor: A divisor or factor of an integer $n$ is an integer $m$ such that there exists an integer $k$ satisfying $n = mk$ . Divisors can be defined in exactly the same way for polynomials or for elements of a commutative ring.

E

Eisenstein: Eisenstein series
Elliptic curve: Elliptic curve
Erdős: Erdős–Kac theorem
Euclid's lemma: Euclid's lemma states that if a prime $p$ divides the product of two integers $ab$ , then $p$ must divide at least one of $a$ or $b$ .
Euler's criterion: Euler's criterion
Euler's theorem: Euler's theorem
Euler's totient function: Euler's totient function

F

factor: See the entry for divisor.
Factorization: Factorization is the process of splitting a mathematical object, often integers or polynomials, into a product of factors.
Fermat's last theorem: Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer $n > 2$ , the equation $a n + b n = c n$ has no positive integer solutions.
Fermat's little theorem: Fermat's little theorem
Fundamental theorem of arithmetic: The fundamental theorem of arithmetic states that every integer greater than 1 can be written uniquely (up to reordering) as a product of primes.

G

Global field: Global field
Goldbach's conjecture: Goldbach's conjecture is a conjecture that states that every even natural number greater than 2 is the sum of two primes.
Greatest common divisor: The greatest common divisor of a finite list of integers is the largest positive number that is a divisor of every integer in the list.

H

Hasse: Hasse's theorem on elliptic curves.
Hecke: Hecke ring

I

ideal: The ideal class group of a number field is the group of fractional ideals in the ring of integers in the field modulo principal ideals. The cardinality of the group is called the class number of the number field. It measures the extent of the failure of unique factorization.
integer: 1. The integers are the numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ .; 2. In algebraic number theory, an integer sometimes means an element of a ring of integers; e.g., a Gaussian integer. To avoid ambiguity, an integer contained in $\mathbb {Q}$ is sometimes called a rational integer.
Iwasawa: Iwasawa theory

L

Langlands: Langlands program
Least common multiple: The least common multiple of a finite list of integers is the smallest positive number that is a multiple of every integer in the list.
local: 1. A local field in number theory is the completion of a number field at a finite place.; 2. The local–global principle.

M

Mersenne prime: A Mersenne prime is a prime number one less than a power of 2.
Modular form: Modular form
modularity theorem: The modularity theorem (which used to be called the Taniyama–Shimura conjecture)

N

number field: A number field, also called an algebraic number field, is a finite-degree field extension of the field of rational numbers.
non-abelian: The non-abelian class field theory is an extension of the class field theory (which is about abelian extensions of number fields) to non-abelian extensions; or at least the idea of such a theory. The non-abelian theory does not exist in a definitive form today.

P

Pell's equation: Pell's equation
place: A place is an equivalence class of non-Archimedean valuations (finite place) or absolute values (infinite place).
prime number: 1. A prime number is a positive integer with no divisors other than itself and 1.; 2. The prime number theorem describes the asymptotic distribution of prime numbers.
profinite: A profinite integer is an element in the profinite completion ${\widehat {\mathbb {Z} }}$ of $\mathbb {Z}$ along all integers.
Pythagorean triple: A Pythagorean triple is three positive integers $a, b, c$ such that $a 2 + b 2 = c 2$ .

R

ramification: The ramification theory.
ring of integers: The ring of integers in a number field is the ring consisting of all algebraic numbers contained in the field.

Q

Quadratic reciprocity: Quadratic reciprocity
Quadratic residue: Quadratic residue

S

Sieve of Eratosthenes: Sieve of Eratosthenes
Square-free integer: A square-free integer is an integer that is not divisible by any square other than 1.
Square number: A square number is an integer that is not the square of an integer.
Szpiro: Szpiro's conjecture is, in a modified form, equivalent to the abc conjecture.

T

Takagi: Takagi existence theorem is a theorem in class field theory.
Twin prime: A twin prime is a prime number that is 2 less or 2 more than another prime number. For example, 7 is a twin prime, since it is prime and 5 is also prime.

V

valuation: valuation (algebra)
valued field: A valued field is a field with a valuation on it.
Vojta: Vojta's conjecture

W

Wilson's theorem: Wilson's theorem states that $n > 1$ is prime if and only if $(n -1)!$ is congruent to $-1$ mod $n$ .

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<span class="mw-page-title-main">Prime number</span> Number divisible only by 1 or itself

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In number theory, the ideal class group of an algebraic number field $K$ is the quotient group $J K / P K$ where $J K$ is the group of fractional ideals of the ring of integers of $K$ , and $P K$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of $K$ . The order of the group, which is finite, is called the class number of $K$ .

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Glossary of number theory

Contents

A

B

C

D

E

F

G

H

I

L

M

N

P

R

Q

S

T

V

W

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