Number Theory Glossary

A complex number that is a root of a monic polynomial with integer coefficients. Generalizes ordinary integers to number fields.

A function defined on positive integers that takes real or complex values, often encoding number-theoretic information. Examples include $\varphi(n)$ and $\mu(n)$.

A measure of how far the ring of integers in an algebraic number field deviates from having unique factorization. A class number of 1 means unique factorization holds.

A positive integer greater than 1 that is not prime, meaning it has at least one positive divisor other than 1 and itself.

A relation between two integers indicating they have the same remainder when divided by a given modulus. Written $a \equiv b \pmod{n}$.

A representation of a number as an integer plus a fraction whose denominator is an integer plus a fraction, and so on. Useful for rational approximation.

Two integers whose greatest common divisor is 1, also called relatively prime.

An infinite series of the form $\sum a_n / n^s$, used extensively in analytic number theory to study the distribution of primes.

A smooth algebraic curve defined by an equation of the form $y^2 = x^3 + ax + b$, used in number theory and cryptography.

A representation of a Dirichlet series as a product over prime numbers, reflecting the multiplicative structure of the integers.

A number of the form $2^{2^n} + 1$. The first five (for $n = 0$ through 4) are prime, but no other Fermat primes are known.

The largest positive integer that divides each of two or more integers without a remainder.

A special subset of a ring that generalizes the concept of divisibility and restores a form of unique factorization in algebraic number theory.

The decomposition of a composite number into a product of smaller integers, ideally primes. Its computational difficulty underpins RSA encryption.

The smallest positive integer that is a multiple of each of two or more integers.

A notation $\left(\frac{a}{p}\right)$ indicating whether integer $a$ is a quadratic residue modulo an odd prime $p$.

The divisor in modular arithmetic that determines the point at which integers 'wrap around.'

An arithmetic function $f$ such that $f(mn) = f(m)f(n)$ whenever $\gcd(m, n) = 1$.

An element of a completion of the rational numbers with respect to a prime $p$, providing an alternative to the real numbers for number-theoretic analysis.

A positive integer that equals the sum of its proper divisors. The first few are 6, 28, and 496. All known perfect numbers are even.

An algorithm that determines whether a given number is prime. Examples include the Miller-Rabin test and the AKS test.

A natural number greater than 1 that has no positive divisors other than 1 and itself.

An integer $a$ is a quadratic residue modulo $n$ if the equation $x^2 \equiv a \pmod{n}$ has a solution.

The function $\zeta(s) = \sum 1/n^s$ for $\text{Re}(s) > 1$, extended by analytic continuation to the whole complex plane except $s = 1$. Central to the study of prime distribution.

A family of techniques for counting or estimating the size of sets of integers satisfying certain divisibility conditions, originating with the Sieve of Eratosthenes.