Integer & Divisor Number Theory¶

← Back to Domain-Specific Families

Abstractions about the structure of positive integers under divisibility and factorization, covering special integer classes (perfect, amicable, highly composite numbers), primality and sieve methods (Miller-Rabin test, Brun sieve), and number-theoretic functions (Mertens function, Dirichlet series, prime-counting function).

45 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

Arithmetic function — A function defined on positive integers, usually with complex values, that encodes a number-theoretic property of each integer.
Arithmetic number — A positive integer whose positive divisors have an integer arithmetic mean.
Brauer's theorem on forms — A theorem guaranteeing large linear spaces of common zeros for sufficiently many-variable homogeneous forms over fields with bounded diagonal-form obstruction.
Brun sieve — A combinatorial sieve that estimates integers avoiding specified prime divisibility conditions by truncating inclusion-exclusion with alternating upper and lower bounds.
Calkin–Wilf tree — A binary tree that enumerates every positive rational number exactly once in lowest terms.
Carmichael number — A composite integer that satisfies the Fermat congruence for every integer base, so it systematically imitates prime behavior under the basic Fermat test.
Class number formula — A number-theoretic identity relating a Dedekind zeta function’s special behavior to a field’s class number, regulator, roots of unity, embeddings, and discriminant.
Composite number — A positive integer greater than one that can be expressed as a product of two smaller positive integers.
Dirichlet density — An analytic density of a set of primes defined by its weighted prime Dirichlet series as the exponent approaches one from above.
Dirichlet series — A complex series whose nth term is a coefficient multiplied by n raised to a complex negative exponent, serving as a multiplicative generating function in analytic number theory.
Dual lattice — The lattice of vectors pairing integrally with every vector of a full-rank lattice under a declared inner product.
Eisenstein reciprocity — A higher-power reciprocity law relating residue symbols in cyclotomic integer rings.
Elementary function arithmetic — A weak first-order arithmetic theory whose provably total functions are the elementary recursive functions, typically extending bounded arithmetic with exponentiation.
Erdős–Woods number — A positive integer k for which some interval of k+1 consecutive integers has every interior member sharing a nontrivial common divisor with at least one endpoint.
Euclid–Mullin sequence — A recursively defined prime sequence taking each new term as the least prime factor of one plus the product of all preceding terms.
Faulhaber's formula — A Bernoulli-number polynomial formula for the sum of equal nonnegative integer powers over the first n positive integers.
Friendly number — A positive integer sharing its abundancy index—the sum of divisors divided by the integer—with at least one distinct positive integer.
Gauss's lemma (number theory) — A criterion computing the Legendre symbol by counting how many least positive residues of a,2a,…,((p−1)/2)a modulo an odd prime exceed p/2.
Generalized taxicab number — The least integer expressible as a sum of a fixed number of positive k-th powers in a specified number of distinct ways.
Hemiperfect number — A positive integer whose sum-of-divisors function divided by the integer is a half-integer with odd numerator.
Highly composite number — A positive integer whose divisor count strictly exceeds that of every smaller positive integer.
Highly powerful number — A powerful integer setting a new record for the product of its prime exponents among all smaller powerful integers.
Highly totient number — An integer whose number of preimages under Euler’s totient function exceeds that of every smaller integer.
Lambek–Moser theorem — A theorem constructing complementary integer sequences from generalized inverse nondecreasing functions.
Lucky number — A natural number surviving an iterative positional sieve that repeatedly deletes every kth remaining number.
Mertens function — The summatory function of the Möbius function, equal to the running difference between square-free integers with even and odd numbers of prime factors.
Miller–Rabin primality test — A randomized strong-probable-prime test that repeatedly checks modular-power witnesses and bounds the chance that a composite integer passes all selected bases.
Modular arithmetic — Arithmetic on congruence classes in which integers differing by a multiple of a fixed positive modulus are identified.
Modular exponentiation — Computation of a power modulo a positive integer, returning the residue of a base raised to an integer exponent without constructing the full power.
Modulus (algebraic number theory) — A finite formal product of places of a global field encoding congruence and ramification conditions for ray class groups and abelian extensions.
Multiplicative partition — An unordered factorization of a positive integer into integers greater than one, with products differing only by factor order identified.
Multiply perfect number — A positive integer whose sum of positive divisors is an integer multiple k of the number itself.
Niven's constant — The limiting average, over positive integers, of the largest exponent in each integer’s prime factorization.
Nonhypotenuse number — A natural number that is not the hypotenuse length of any integer-sided right triangle.
Odious number — A nonnegative integer whose binary expansion contains an odd number of one bits.
P-adic number — An element of the completion of the rational numbers under the non-Archimedean absolute value determined by a prime p.
Prime triplet — A set of three prime numbers spanning six integers, necessarily in one of two offset patterns apart from exceptional triples containing three.
Prime-counting function — The arithmetic function pi of x that counts prime numbers less than or equal to a real bound x.
Square number — An integer equal to the product of some integer with itself.
Størmer number — A positive integer whose squared value plus one has a prime factor at least twice the original integer.
Super-Poulet number — A composite base-two pseudoprime for which every positive divisor d also divides two-to-the-d minus two.
Supernatural number — A formal prime product whose exponent at each prime is a natural number or infinity, extending positive integers under divisibility.
Tree of primitive Pythagorean triples — A rooted ternary tree that generates every primitive positive integer solution of the Pythagorean equation exactly once by fixed linear transformations.
Unusual number — A natural number whose largest prime factor is strictly greater than its square root.
Woodall number — A natural number of the form n times two to the n minus one.