Number Theory¶

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65 domain-specific abstractions whose origin domain is Number Theory.

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.
Auxiliary function — A deliberately constructed function with engineered zeros, growth or arithmetic properties that converts a target claim—especially in transcendence theory—into an estimate or contradiction.
Calkin–Wilf tree — A binary tree that enumerates every positive rational number exactly once in lowest terms.
Champernowne constant — The real number formed by concatenating the positive integers in order in a chosen base, with the base-ten version proved normal.
Cullen number — An integer of the form C_n = n·2^n + 1, forming a named exponential sequence whose rare prime terms are Cullen primes.
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.
Dedekind psi function — The multiplicative arithmetic function ψ(n)=n times the product of (1+1/p) over the distinct prime divisors of n.
Deficient number — Classify a positive integer as deficient when the sum of its positive proper divisors is smaller than the integer itself, equivalently when its divisor sum is less than twice the integer.
Diophantine Equation — A polynomial equation with integer coefficients whose admissible solutions are required to be integers.
Diophantine quintuple — A five-element set of positive integers for which the product of every two distinct elements plus one is a perfect square.
Divisor Function — The multiplicative arithmetic-function family \(\sigma_z(n)=\sum_{d\mid n}d^z\), including divisor count and divisor sum as distinguished cases.
Equidistributed sequence — Require the limiting frequency of sequence terms in every subinterval to equal that subinterval’s normalized length.
Euclid number — An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.
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.
Euler's totient function — The arithmetic function counting residue classes modulo a positive integer that are coprime to it.
Fermat number — Generate the integer sequence F_n = 2⁽²n) + 1, whose product recurrence makes distinct terms pairwise coprime and whose rare prime members connect to constructible polygons.
Fortunate number — For each positive index n, select the least integer m greater than one for which the nth primorial plus m is prime, producing the sequence governed by Fortune's still-open primality conjecture.
Friendly number — A positive integer sharing its abundancy index—the sum of divisors divided by the integer—with at least one distinct positive integer.
Fundamental Theorem of Arithmetic — The integer-specific theorem that every integer greater than one is a product of primes and that its prime multiset is unique up to order.
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 cototient number — A positive integer k>1 having more solutions to x−φ(x)=k than any smaller integer greater than one, where φ is Euler's totient function.
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.
Integer factorization — The decomposition of a positive integer into integer factors, canonically into a unique multiset of primes up to ordering, with computational difficulty depending strongly on input size and structure.
Kaprekar number — Classify a base-b natural number whose square can be split at a declared digit position into two parts that sum back to the number.
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.
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.
Modular arithmetic — Arithmetic on congruence classes in which integers differing by a multiple of a fixed positive modulus are identified.
Multiplicative Digital Root — The terminal single base-b digit reached by repeatedly replacing a nonnegative integer with the product of its digits, paired with multiplicative persistence as the number of iterations required to reach that fixed point.
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.
Möbius function — The multiplicative arithmetic function μ(n) that is zero on numbers divisible by a prime square and otherwise equals minus one to the number of distinct prime factors.
Noncototient — A positive integer that is not equal to n−φ(n) for any positive integer n, where φ is Euler's totient function.
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.
Partition function (number theory) — The arithmetic function p(n) that counts unordered representations of a nonnegative integer as a sum of positive integers, with generating-function, recurrence, asymptotic and modular-congruence structure.
Perfect number — Classify a positive integer as perfect when the sum of all its proper positive divisors equals the integer itself, equivalently when its divisor-sum is exactly twice the integer.
Perrin number — A doubly infinite integer sequence generated from initial values 3, 0 and 2 by adding terms two and three positions earlier.
Pillai's Arithmetical Function — The multiplicative gcd-sum function P(n)=sum from k=1 to n of gcd(k,n), whose divisor-class decomposition P=id*phi exposes prime-power evaluation, Euler products, and average-order analysis.
Prime signature — Classify a positive integer by the unordered multiset of positive exponents in its unique prime factorization, discarding prime labels while preserving multiplicative shape.
Prime triplet — A set of three prime numbers spanning six integers, necessarily in one of two offset patterns apart from exceptional triples containing three.
Primefree Sequence — A nontrivial Fibonacci-type integer sequence begun from coprime composite seeds and proved to contain only composite terms, typically by a finite cover of periodic modular divisibility classes.
Prouhet–Thue–Morse constant — The real number whose binary expansion is the Thue–Morse sequence.
Quasiperfect number — Classify a positive integer by the divisor-sum equation sigma of n equals twice n plus one, an unresolved class whose hypothetical members must satisfy strong odd-square and prime-factor restrictions.
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.
Reduced residue system — A complete set of incongruent representatives modulo n chosen from the integer classes coprime to n.
Sparsely Totient Number — A natural number n whose Euler totient is a strict suffix minimum: every larger integer m has φ(m) greater than φ(n).
Størmer number — A positive integer whose squared value plus one has a prime factor at least twice the original integer.
Sublime number — A positive integer having a perfect number of positive divisors and a divisor sum that is itself a perfect number.
Sum of squares function — Count ordered signed integer k-tuples whose squared coordinates sum to n, yielding the arithmetic function r_k(n) and its divisor-sum, theta-series, and local-obstruction structure.
Sum-Free Sequence — A strictly increasing sequence of positive integers in which no term is representable as a sum of a subset of its predecessors, coupling prefix-dependent additive avoidance to sparse-growth and reciprocal-sum questions.
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.
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.
Tunnell's theorem — Test a square-free integer for the congruent-number property through equalities among counts of representations by four ternary quadratic forms—necessary unconditionally and sufficient conditional on Birch–Swinnerton-Dyer.
Unusual number — A natural number whose largest prime factor is strictly greater than its square root.
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.
Woodall number — A natural number of the form n times two to the n minus one.