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Length of expression as combination of 1s
In number theory, the complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions,
Integer_complexity
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Inherent difficulty of computational problems
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource
Computational complexity theory
Computational_complexity_theory
Algorithmic runtime requirements for common math procedures
stands in for the complexity of the chosen multiplication algorithm. This table lists the complexity of mathematical operations on integers. On stronger computational
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Number in {..., –2, –1, 0, 1, 2, ...}
factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Amount of resources to perform an algorithm
arithmetic complexity. For example, the arithmetic complexity of the computation of the determinant of a n×n integer matrix is O ( n 3 ) {\displaystyle O(n^{3})}
Computational_complexity
Complexity class used to classify decision problems
problems in computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems
NP_(complexity)
Method to solve optimization problems
H. Freeman. ISBN 978-0-7167-1045-5. A6: MP1: INTEGER PROGRAMMING, pg.245. (computer science, complexity theory) Gärtner, Bernd; Matoušek, Jiří (2006)
Linear_programming
Estimate of time taken for running an algorithm
the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly
Time_complexity
Measure of algorithmic complexity
determines whether the nth proof actually proves a complexity formula K(s) ≥ L. The strings s, and the integer L in turn, are computable by procedure: function
Kolmogorov_complexity
Algorithmic runtime requirements for matrix multiplication
complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix"
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Unsolved problem in computer science
axioms for integer arithmetic, then nearly polynomial-time algorithms exist for all NP problems. Therefore, assuming (as most complexity theorists do)
P_versus_NP_problem
Topics referred to by the same term
01 Complexity (journal) Computational complexity, of algorithms Computational complexity theory Game complexity, in combinatorial game theory Integer complexity
Complexity_(disambiguation)
Estimate of number of possible chess games
efficiently computable bijection between integers and chess positions. Allis also estimated the game-tree complexity to be at least 10123, "based on an average
Shannon_number
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Algorithm to multiply two numbers
n} -bit integers. This is known as the computational complexity of multiplication. Usual algorithms done by hand have asymptotic complexity of O ( n
Multiplication_algorithm
Quantum algorithm for integer factorization
can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers can be done with a polynomial complexity circuit on an
Shor's_algorithm
Measure of complexity of real-valued functions
learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with
Rademacher_complexity
Largest integer that divides given integers
of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest
Greatest_common_divisor
polyhedron with facet complexity at most f, and v is a rational vector with distance from P of at most 2-6nf, and q and w are integer vectors satisfying
N-dimensional_polyhedron
Computation modulo a fixed integer
mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching
Modular_arithmetic
Typographic symbol
truncation of A {\textstyle A} ") Integer complexity: ‖ n ‖ {\displaystyle \|n\|} ; reads "the complexity of the integer n". In LaTeX mathematical mode,
Vertical_bar
Calculations where numbers' precision is only limited by computer memory
for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Class of problems in computer science
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability
PP_(complexity)
Algorithm that arranges lists in order
better than O(n log n) time complexity assuming certain constraints, including: Thorup's algorithm, a randomized integer sorting algorithm, taking O(n
Sorting_algorithm
Branch of computational complexity theory
parameterized complexity was fixed-parameter tractability. Many problems have the following form: given an object x and a nonnegative integer k, does x have
Parameterized_complexity
Model of computation
problem. If the input is an integer circuit, however, it is unknown whether this problem is decidable. Circuit complexity attempts to classify Boolean
Circuit_(computer_science)
Approach to the study of finite semigroups and automata
semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example
Krohn–Rhodes_theory
Open problem on 3x+1 and x/2 functions
"Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability. 1 (1): 19–56. doi:10.3233/COM-150032. Michel
Collatz_conjecture
Greatest integer less than or equal to square root
number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal
Integer_square_root
Decidable first-order theory of the natural numbers with addition
structure of non-negative integers with constants 0 {\displaystyle 0} , 1 {\displaystyle 1} , and the addition of non-negative integers. Presburger arithmetic
Presburger_arithmetic
Problem a computer might be able to solve
a positive integer n, find a nontrivial prime factor of n." is a computational problem that has a solution, as there are many known integer factorization
Computational_problem
Class in computational complexity theory
}{=}}{\mathsf {P}}} More unsolved problems in computer science In computational complexity theory, the class NC (for "Nick's Class") is the set of decision problems
NC_(complexity)
Self-referential paradox
self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters)
Berry_paradox
Combinational digital circuit
combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers. This is in contrast to a floating-point unit (FPU), which
Arithmetic_logic_unit
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A003037 (Smallest number of complexity n: smallest number requiring
1000_(number)
Graver bases enable iterative solutions of linear and various nonlinear integer programming problems in polynomial time. They were introduced by Jack E
Graver_basis
Problem of inverting exponentiation in groups
generalizes this concept to a cyclic group. A simple example is the group of integers modulo a prime number (such as 5) under modular multiplication of nonzero
Discrete_logarithm
Algorithm that employs a degree of randomness as part of its logic or procedure
Carlo algorithms are considered, and several complexity classes are studied. The most basic randomized complexity class is RP, which is the class of decision
Randomized_algorithm
2002 documentary film by Jeffrey Blitz
a PhD in Mathematics from the University of Michigan, focusing on integer complexity. As of 2025, Altman is involved in the New York rationalist community
Spellbound_(2002_film)
Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
computational complexity theory, an integer circuit is a circuit model of computation in which inputs to the circuit are sets of integers and each gate
Integer_circuit
Model of computational complexity
learning algorithms. Many circuit complexity classes are defined in terms of class hierarchies. For each non-negative integer i, there is a class NCi, consisting
Circuit_complexity
Venezuelan computer scientist
recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking". Blum
Manuel_Blum
Type of computer science algorithm
that space complexity also has varied choices in whether or not to count the index lengths as part of the space used. Often, the space complexity is given
In-place_algorithm
Central computer component that executes instructions
encoded integer) that the CPU can process in one operation, which is commonly called word size, bit width, data path width, integer precision, or integer size
Central_processing_unit
Product of the prime factors of an integer
Encyclopedia of Integer Sequences. OEIS Foundation. Adleman, Leonard M.; McCurley, Kevin S. (1994). "Open Problems in Number Theoretic Complexity, II". Algorithmic
Radical_of_an_integer
Complexity class of bounded-depth circuits
representations of integers). Since it is a circuit class, like P/poly, AC0 also contains every unary language. From a descriptive complexity viewpoint, DLOGTIME-uniform
AC0
Decision problem in computer science
multiple subset sum problem. The time complexity of SSP depends on two parameters: n - the number of input integers. If n is a small fixed number, then
Subset_sum_problem
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with
List of computability and complexity topics
List_of_computability_and_complexity_topics
Exponent of a power of two
The number of digits (bits) in the binary representation of a positive integer n is the integral part of 1 + log2 n, i.e. ⌊ log 2 n ⌋ + 1. {\displaystyle
Binary_logarithm
Product of numbers from 1 to n
factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers less than or equal to
Factorial
Study of resources used by an algorithm
the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to
Analysis_of_algorithms
Algorithm for determinants of integers
Bareiss algorithm is not commonly used for integer matrices, because multi-modular arithmetic allows a complexity similar to that of the Bareiss algorithm
Bareiss_algorithm
Associative array for storing key–value pairs
probing sequence. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the
Hash_table
Date and time representation system widely used in computing
referred to as the Unix epoch. Unix time is typically encoded as a signed integer. The Unix time 0 is exactly midnight UTC on 1 January 1970, with Unix time
Unix_time
Algorithm to compute the maximum flow in a network
found in O ( E ) {\displaystyle O(E)} time and increases the flow by an integer amount of at least 1 {\displaystyle 1} , with the upper bound f {\displaystyle
Ford–Fulkerson_algorithm
Ancient algorithm for generating prime numbers
numbers less than or equal to a given integer n by Eratosthenes's method: Create a list of consecutive integers from 2 through n: (2, 3, 4, ..., n). Initially
Sieve_of_Eratosthenes
Mathematical theorem
the integration formulas for integrand's with and without consecutive integer exponents and for single and double integrals. The integration formula
Ramanujan's_master_theorem
Arithmetical operation
Joris van der Hoeven submitted a paper presenting an integer multiplication algorithm with a complexity of O ( n log n ) . {\displaystyle O(n\log n).} The
Multiplication
Algorithm for generating prime numbers
is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes, which marks off multiples
Sieve_of_Atkin
Complexity class used in circuit complexity
numbers, integer division or recognizing the Dyck language with multiple types of parentheses. It is commonly used to model the computational complexity of
TC0
Function that counts distinct factors of a string
of a positive integer n to be the number of different words of length n in L The complexity function of a word is thus the complexity function of the
Complexity_function
Interactive proof system in computational complexity theory
In computational complexity theory, an Arthur–Merlin protocol, introduced by Babai (1985), is an interactive proof system in which the verifier's coin
Arthur–Merlin_protocol
Subfield of mathematical optimization
is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial
Combinatorial_optimization
Computational complexity of quantum algorithms
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational
Quantum_complexity_theory
Complexity class
In computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time
NP-hardness
Attribute of machine learning models
The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function
Sample_complexity
Type of mathematical expression
addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of
Polynomial
Discrete Fourier transform algorithm
of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O ( n 2 ) {\textstyle O(n^{2})} , which arises
Fast_Fourier_transform
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently
Complement_(complexity)
problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms
Weak_NP-completeness
Complexity class
problem to unconditionally show that integer factorization is in PPP. Christos Papadimitriou (1994). "On the complexity of the parity argument and other inefficient
PPP_(complexity)
Algorithm for generating prime numbers
up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934. The sieve starts with a list of the integers from 1 to n. From
Sieve_of_Sundaram
Algorithm checking for prime numbers
test is based upon the following theorem: Given an integer n ≥ 2 {\displaystyle n\geq 2} and integer a {\displaystyle a} coprime to n {\displaystyle n}
AKS_primality_test
Algorithm for computing logarithms
discrete logarithms in a finite abelian group whose order is a smooth integer. The algorithm was introduced by Roland Silver, but first published by
Pohlig–Hellman_algorithm
Complexity class
computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class
Co-NP
PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided in time bounded by
PR_(complexity)
Two-dimensional packing problem
an integer, the answer is a 2 , {\displaystyle a^{2},} but the precise – or even asymptotic – amount of unfilled space for an arbitrary non-integer a {\displaystyle
Square_packing
Used to count, measure, and label
Hindu–Arabic numeral system, a decimal system which can display any non-negative integer using a combination of ten Arabic numeral symbols called digits. Numerals
Number
Probabilistic primality test
{\displaystyle s} is a positive integer and d {\displaystyle d} is an odd positive integer. Let’s consider an integer a {\displaystyle a} , called a
Miller–Rabin_primality_test
Efficient sorting algorithm that combines insert and merge operations
= numerator = 0 while (integer_part < array.size) // get the ranges for A and B start = integer_part integer_part += integer_step numerator += numerator_step
Block_sort
Divide and conquer sorting algorithm
"Algorithms and Complexity". Proceedings of the 3rd Italian Conference on Algorithms and Complexity. Italian Conference on Algorithms and Complexity. Lecture
Merge_sort
Algorithm in computational number theory
{b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Non-comparative lexicographical sorting algorithm
sort can be applied to data that can be sorted lexicographically, be they integers, words, punch cards, playing cards, or the mail. Radix sort dates back
Radix_sort
Class of binary number
On-Line Encyclopedia of Integer Sequences, OEIS Foundation Charlier, Émilie; Cisternino, Célia; Massuir, Adeline (2019), "State complexity of the multiples of
Evil_number
Search algorithm finding the position of a target value within a sorted array
floor of R − L 2 {\displaystyle {\frac {R-L}{2}}} , which is the greatest integer less than or equal to R − L 2 {\displaystyle {\frac {R-L}{2}}} . If A m
Binary_search
different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle
State_complexity
Natural sorting algorithm
representing the positive integer 3 (because they each contain three beads) while the top row represents the positive integer 2 (as it only contains two
Bead_sort
Even integers as sums of two primes
Euler (letter XLIII), in which he proposed the following conjecture: Every integer that can be written as the sum of two primes can also be written as the
Goldbach's_conjecture
Search tree data structure
trie, which uses individual bits from fixed-length binary data (such as integers or memory addresses) as keys. The idea of a trie for representing a set
Trie
Algorithms to generate prime numbers
sieves are most common. A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which
Generation_of_primes
Number whose square is a given number
roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they
Square_root
of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics
List_of_complexity_classes
Problem in computer science
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose
Set_packing
Complexity class
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution
PLS_(complexity)
Type qualifier denoting the data as being read-only
j; } }; void foo(Integer& nonConstInteger, const Integer& constInteger) { int y = nonConstInteger.get(); // OK int x = constInteger.get(); // OK: get()
Const_(computer_programming)
Microarchitecture by AMD
CPU. In terms of hardware complexity and functionality, a Bulldozer CMT module is equal to a dual-core processor in its integer calculation capabilities
Bulldozer_(microarchitecture)
INTEGER COMPLEXITY
INTEGER COMPLEXITY
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Boy/Male
Norse
Son's army.
Boy/Male
Muslim
To wait
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Boy/Male
Arabic, Muslim
To Wait
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
INTEGER COMPLEXITY
INTEGER COMPLEXITY
Female
Japanese
(å¤ç¾Ž) Japanese name NATSUMI means "summer beauty."
Boy/Male
British, English
From the Enclosed Meadow
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Scholar
Female
French
Variant spelling of French Alaina, possibly ALAYNA means "little rock."Â
Female
Persian/Iranian
Persian name derived from the word nîgâh, literally NÎGA means "sight, vision," but usually rendered "looking after."
Girl/Female
Indian
Fame; Succesful; Popular
Boy/Male
Egyptian
Name of a prince.
Girl/Female
Arabic
Prophet Name
Boy/Male
English
Maker of rope.
Boy/Male
Arabic, Muslim
Character
INTEGER COMPLEXITY
INTEGER COMPLEXITY
INTEGER COMPLEXITY
INTEGER COMPLEXITY
INTEGER COMPLEXITY
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
p. pr. & vb. n.
of Inter
v. t.
To inter again.
v. t.
To inter.
v. t.
To deposit or inter in a chapel; to enshrine.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
v. t.
To inhume; to bury; to inter.
imp. & p. p.
of Inter
n.
One who inters.
n.
One who makes an index.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
n.
One who intends.
n.
One who makes an entrance or beginning.
v. t.
To inter with funeral rites; to bury.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
n.
One who gathers the vintage.
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
v. t.
To place in a tomb; to bury; to inter; to entomb.