Algebraic Number Theory & Reciprocity

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Abstractions about algebraic and modular properties of numbers and fields. They include class formulas, residues and reciprocity laws, local fields, Diophantine configurations, modular inverses, arithmetic functions, special integer sequences, and unresolved number-theoretic conjectures.

28 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.

• Bhargava factorial — A factorial sequence attached to an arbitrary subset of the integers through p-orderings, generalizing n-factorial while preserving divisibility and integer-valued-polynomial properties.
• Biquadratic field — A degree-four Galois extension of the rational numbers with Klein-four Galois group.
• 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.
• Cannonball problem — The Diophantine problem of finding integers for which a square pyramidal number is also a perfect square, equivalently when the sum of consecutive squares from one to n is itself square.
• 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.
• Cunningham number — An integer of the form b^n−1 or b^n+1 with integer base b that is not itself a perfect power, organizing prominent exponential families for factorization and primality study.
• Different ideal — An ideal measuring the failure of the ring of integers of a number field to be self-dual under the trace pairing, inverse to the codifferent fractional ideal.
• Diophantine quintuple — A five-element set of positive integers for which the product of every two distinct elements plus one is a perfect square.
• 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.
• Golden field — The real quadratic number field obtained by adjoining the square root of five to the rationals.
• Knödel number — For a fixed positive integer n, a composite integer m such that a^(m−n) is congruent to one modulo m for every integer a coprime to m.
• Littlewood conjecture — The open conjecture that every pair of real numbers admits arbitrarily strong simultaneous rational approximation with a common denominator in a multiplicative sense.
• Local class field theory — The theory that classifies finite abelian extensions of a local field through a reciprocity map from the field's multiplicative group to its abelian Galois group.
• Local field — A nondiscrete locally compact Hausdorff topological field, equivalently in the non-Archimedean case a complete discretely valued field with finite residue field, serving as a completion-scale model of global arithmetic.
• Mahler measure — A multiplicative height-like measure of a polynomial equal to its leading coefficient magnitude times the moduli of roots outside the unit circle.
• Modular arithmetic — Arithmetic on congruence classes in which integers differing by a multiple of a fixed positive modulus are identified.
• Modular multiplicative inverse — A residue x whose product with a given integer a is congruent to one modulo m, existing exactly when a and m are coprime.
• 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.
• Perrin number — A doubly infinite integer sequence generated from initial values 3, 0 and 2 by adding terms two and three positions earlier.
• Quadratic residuosity problem — The computational decision problem of determining whether a number with Jacobi symbol one is a square modulo a composite whose prime factorization is unknown.
• Ramanujan's sum — The finite exponential sum c_q(n) over residues coprime to q, an integer-valued arithmetic function used as a Fourier basis for number-theoretic expansions.
• Reciprocity law — A number-theoretic rule relating splitting behavior of primes in an algebraic extension to congruence or residue information, generalizing quadratic reciprocity.
• Reduced residue system — A complete set of incongruent representatives modulo n chosen from the integer classes coprime to n.
• Thabit number — An integer of the form 3·2^n−1 for a nonnegative integer n, historically linked through special primality conditions to constructions of amicable numbers.
• 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.
• Wall–Sun–Sun prime — A conjectural prime p for which p² divides the Fibonacci number indexed by p's Pisano period, equivalently a Fibonacci–Wieferich prime under standard formulations.
• Wilson quotient — For a prime p, the integer ((p−1)!+1)/p, whose residues encode refinements of Wilson's theorem and define Wilson primes when divisible by p.