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Greatest integer less than or equal to square root
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 to the square root
Integer_square_root
Algorithms for calculating square roots
In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may
Square_root_algorithms
Number whose square is a given number
the square root of numbers having many digits. It was known to the ancient Greeks that square roots of positive integers that are not perfect squares are
Square_root
Product of an integer with itself
real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, 9 =
Square_number
Root-finding algorithm
approximation through integer operations by adding and subtracting the integer form of floating-point numbers, and taking a square root by dividing by two
Fast_inverse_square_root
Unique positive real number which when multiplied by itself gives 2
A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 30 decimal
Square_root_of_2
Positive real number which when multiplied by itself gives 7
expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Robert Nemiroff; Jerry Bonnell (2008). The square root of 7.
Square_root_of_7
Irrational algebraic number
In mathematics, the square root of 10 is the positive real number that, when multiplied by itself, gives the number 10. It is approximately equal to 3
Square_root_of_10
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
Complex number whose mapping on a coordinate plane produces a triangular lattice
cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice
Eisenstein_integer
Integer factorization algorithm
25 because the square of the next prime is 49, and below n = 25 just 2 and 3 are sufficient. Should the square root of n be an integer, then it is a factor
Trial_division
Mathematical procedure
An integer relation between a set of real numbers x1, x2, ..., xn is a set of integers a1, a2, ..., an, not all 0, such that a 1 x 1 + a 2 x 2 + ⋯ + a
Integer_relation_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
Principal square root of minus 1
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Nahin, Paul J. (1998). An Imaginary Tale: The story of i [the square root of minus one]. Chichester:
Imaginary_unit
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
Unique positive real number which when multiplied by itself gives 3
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3 {\textstyle {\sqrt
Square_root_of_3
Root of a quadratic polynomial with a unit leading coefficient
quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root of some
Quadratic_integer
Lattice group in Euclidean space whose points are integer n-tuples
n-tuples of integers. The two-dimensional integer lattice is also called the square lattice (or grid lattice) and the three-dimensional integer lattice is
Integer_lattice
Complex number that solves a monic polynomial with integer coefficients
theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial
Algebraic_integer
Positive real number which when multiplied by itself gives 5
The square root of 5, denoted 5 {\displaystyle {\sqrt {5}}} , is the positive real number that, when multiplied by itself, gives the natural number
Square_root_of_5
Two-dimensional packing problem
half-integer, the wasted space is at least proportional to its square root. The precise asymptotic growth rate of the wasted space, even for half-integer side
Square_packing
Exponentation in modular arithmetic
is the remainder c when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c
Modular_exponentiation
Method for computing the relation of two integers with their greatest common divisor
the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that a x + b y = gcd (
Extended_Euclidean_algorithm
Algorithm in computational number theory
believed that r=1.618034 is a (slightly rounded) root to an unknown quadratic equation with integer coefficients, one may apply LLL reduction to the lattice
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Number with an integer power equal to 1
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power n. Roots of unity are used in many branches
Root_of_unity
Mathematical proof technique
jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with
Vieta_jumping
Mathematical operation
mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix
Square_root_of_a_matrix
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
Algorithm for computing greatest common divisors
polynomials. The Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers and i is the square root of negative one. By
Euclidean_algorithm
Positive real number which when multiplied by itself gives 6
(ed.). "Sequence A010464 (Decimal expansion of square root of 6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Robert Nemiroff; Jerry
Square_root_of_6
System of rapid mental calculation
subtraction and square root." (1960) "The best selling method for high-speed multiplication, division, addition, subtraction and square root – without a calculator
Trachtenberg_system
Problem of inverting exponentiation in groups
algorithms for integer factorization. These algorithms run faster than the naïve algorithm, some of them proportional to the square root of the size of
Discrete_logarithm
Method for division with remainder
A division algorithm is an algorithm which, given two integers N and D (respectively the numerator and the denominator), computes their quotient and/or
Division_algorithm
Algorithm to multiply two numbers
squares, takes the difference of the results, and divides by four by shifting two bits to the right. For 8-bit integers the table of quarter squares will
Multiplication_algorithm
Modular arithmetic concept
root modulo n if every number a coprime to n is congruent to a power of g modulo n. In symbols, g is a primitive root modulo n if for every integer a
Primitive_root_modulo_n
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
Number in {..., –2, –1, 0, 1, 2, ...}
example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, 5/4, and the square root of 2 are not. The integers form the smallest group and the smallest
Integer
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
Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
Arithmetic operation, inverse of nth power
positive integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and
Nth_root
Natural number
The chemical element with atomic number 2 is helium. Binary number Square root of 2 −2 Colman, Samuel (1912). Coan, C. Arthur (ed.). Nature's Harmonic
2
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
Natural number
normal magic square, called the Luoshu square. All integers n ≥ 34 {\displaystyle n\geq 34} can be expressed as the sum of five non-zero squares. There are
5
Factorization algorithm
these homomorphisms will map each "square root" (typically not represented as a rational number) into its integer representative. Now the product of the
General_number_field_sieve
Number that is not a ratio of integers
irrationality of the square root of two can be generalized using the fundamental theorem of arithmetic. This asserts that every integer has a unique factorization
Irrational_number
Geometric arrangements of points, foundational to Lie theory
where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one
Root_system
Arithmetic operation
apply the square super-root twice: x = s s r t ( s s r t ( y x ) ) {\displaystyle x=\mathrm {ssrt} (\mathrm {ssrt} (y^{x}))} . For each integer n > 2, the
Tetration
Figurate number
formula. So an integer x is triangular if and only if 8x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the
Triangular_number
Standard division algorithm for multi-digit numbers
digit of the dividend by the divisor. The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated
Long_division
Algorithm for finding zeros of functions
Aitken's delta-squared process Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem
Newton's_method
(except possibly at 0). Integer-valued functions defined on the domain of non-negative real numbers include the integer square root function and the prime-counting
Integer-valued_function
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
System that regulates the formation of blocks on a blockchain
usage scenario. Here is a list of known proof-of-work functions: Integer square root modulo a large prime[dubious – discuss] Weaken Fiat–Shamir signatures
Proof_of_work
Algorithm for solving the discrete logarithm problem
and a group element β {\displaystyle \beta } , the problem is to find an integer x {\displaystyle x} such that α x = β . {\displaystyle \alpha ^{x}=\beta
Baby-step_giant-step
Computational method
has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex
Factorization_of_polynomials
Algorithm for checking if a number is prime
concise verification that n is prime. Let n be a positive integer. If there exists an integer a, 1 < a < n, such that a n − 1 ≡ 1 ( mod n ) {\displaystyle
Lucas_primality_test
Integer that is both a perfect square and a triangular number
sum of all integers from 1 {\displaystyle 1} to n {\displaystyle n} has a square root that is an integer. There are infinitely many square triangular
Square_triangular_number
Product of two distinct primes ≡ 3 (mod 4)
Given n = p × q a Blum integer, Qn the set of all quadratic residues modulo n and coprime to n and a ∈ Qn. Then: a has four square roots modulo n, exactly
Blum_integer
Factorization method based on the difference of two squares
Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference
Fermat's_factorization_method
Probabilistic primality testing algorithm
bounty on a strengthened version of this test. Let n be the odd positive integer that we wish to test for primality. Optionally, perform trial division
Baillie–PSW_primality_test
Product of a number by itself
to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In algebra, the operation of squaring is often
Square_(algebra)
Every natural number can be represented as the sum of four integer squares
four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative integer squares
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Quantum algorithm for integer factorization
Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. It
Shor's_algorithm
Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Other algorithms Chakravala Cornacchia Exponentiation by squaring Integer square root Integer
Korkine–Zolotarev lattice basis reduction algorithm
Korkine–Zolotarev_lattice_basis_reduction_algorithm
numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. Transfinite numbers: Numbers that are
List_of_types_of_numbers
Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number
List_of_number_theory_topics
Integer factorization algorithm
linear sieve. The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n
Quadratic_sieve
Function that, applied twice, gives another function
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition
Functional_square_root
Algebraic construction
ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n +
Ring_of_integers
Test if a Mersenne number is prime
{\displaystyle \omega ^{2^{p-2}}+{\bar {\omega }}^{2^{p-2}}=kM_{p}} for some integer k, so ω 2 p − 2 = k M p − ω ¯ 2 p − 2 . {\displaystyle \omega ^{2^{p-2}}=kM_{p}-{\bar
Lucas–Lehmer_primality_test
Mathematical concept
irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct
Quadratic_irrational_number
Natural number
square of six, and the eighth triangular number or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square
36_(number)
One of several equivalent definitions of a computable function
more complicated way, since they are all primitive recursive. The integer square root of x can be defined as the least z such that ( z + 1 ) 2 > x {\displaystyle
General_recursive_function
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
Arithmetic operation
numbers: the base, b, and the exponent or power, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that
Exponentiation
Algorithm used in modular arithmetic
prime: that is, to find a square root of n modulo p. The Tonelli–Shanks algorithm cannot be used for composite moduli: finding square roots modulo composite
Tonelli–Shanks_algorithm
Probabilistic primality test
{p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds. If it does not
Fermat_primality_test
Multiplication algorithm
integers, published by Arnold Schönhage and Volker Strassen in 1971. It works by recursively applying fast Fourier transform (FFT) over the integers modulo
Schönhage–Strassen_algorithm
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed
1,000,000
Square root of a non-positive real number
Alexandria is noted as the first to present a calculation involving the square root of a negative number, it was Rafael Bombelli who first set down the rules
Imaginary_number
Natural number
the sum of three cubes. There are nine Heegner numbers, or square-free positive integers n {\displaystyle n} that yield an imaginary quadratic field
9
Algorithm for computing the greatest common divisor
algorithm that computes the greatest common divisor (GCD) of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional
Binary_GCD_algorithm
Number-theoretic algorithm
{m-r_{k}^{2}}{d}}}} is an integer, then the solution is x = r k , y = s {\displaystyle x=r_{k},y=s} ; otherwise try another root of -d until either a solution
Cornacchia's_algorithm
Integer factorization algorithm
amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized
Pollard's_rho_algorithm
Condition under which an odd prime is a sum of two squares
sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2},} with x and y integers, if and only
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Probabilistic primality test
RSA cryptosystem. Euler proved that for any odd prime number p and any integer a, a ( p − 1 ) / 2 ≡ ( a p ) ( mod p ) {\displaystyle a^{(p-1)/2}\equiv
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Problem in computer science
Turing run-time complexity of the square-root sum problem? More unsolved problems in computer science The square-root sum problem (SRS) is a computational
Square-root_sum_problem
Mathematical algorithm
discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute γ {\displaystyle \gamma }
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
Special-purpose integer factorization algorithm
integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of
Special_number_field_sieve
Type of complex number
mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For
Algebraic_number
Natural number
consequence of the fact that 41 is a factor of 99999. the smallest integer whose square root has a simple continued fraction with period 3. a prime index prime
41_(number)
Algorithm for generating numbers coprime with first few primes
method can thus be used for an improvement of the trial division method for integer factorization, as none of the generated numbers need be tested in trial
Wheel_factorization
Mathematical for factoring integers
squares in Fermat's factorization method. The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer
Euler's_factorization_method
Natural number, composite number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-18. Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)".
14_(number)
Probabilistic algorithm for computing discrete logarithms
← empty_list for k = 1 , 2 , … {\displaystyle k=1,2,\ldots } Using an integer factorization algorithm optimized for smooth numbers, try to factor g k
Index_calculus_algorithm
Algorithm in computational number theory
pseudorandom map f : G → S {\displaystyle f:G\rightarrow S} . 2. Choose an integer N {\displaystyle N} and compute a sequence of group elements { x 0 , x
Pollard's_kangaroo_algorithm
Algorithm for multiplying large numbers
description of it, is a multiplication algorithm for large integers. Given two large integers, a and b, Toom–Cook splits up a and b into k smaller parts
Toom–Cook_multiplication
Rational number equal to an integer plus 1/2
{n}{2}}+1)}}R^{n}~.} The values of the gamma function on half-integers are rational multiples of the square root of pi: Γ ( 1 2 + n ) = ( 2 n − 1 ) ! ! 2 n π
Half-integer
Natural number
104 or 1 E+4 (equivalently 1 E4) in E notation. It is the square of 100 and the square root of 100,000,000. The value of a myriad to the power of itself
10,000
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A140480 (RMS numbers: numbers n such that root mean square of divisors
1000_(number)
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Boy/Male
English American
Shieldbearer.
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Surname or Lastname
English
English : patronymic from Squire.
Boy/Male
English
Shieldbearer.
Boy/Male
Muslim
To wait
Surname or Lastname
English
English : variant of Spear.
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Boy/Male
French Latin
A squire.
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
Boy/Male
Arabic, Muslim
To Wait
Boy/Male
American, British, English
Shield Bearer
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Surname or Lastname
English
English : variant of Squire.
Boy/Male
Italian
Squire.
Boy/Male
British, English
Spear-man
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Boy/Male
Anglo Saxon American English Scottish
Steward.
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
Boy/Male
Tamil
Harshaman | ஹரà¯à®·à®®à®¨
Full of Joy
Boy/Male
Tamil
Samudragupta | ஸமà¯à®¤à¯à®°à®•à¯à®ªà¯à®¤à®¾Â
A famous gupta king
Surname or Lastname
English
English : variant spelling of Tennyson.
Female
Egyptian
, the mother of prince Namurot.
Girl/Female
American, Australian, British, Christian, Danish, Dutch, English, Finnish, German, Greek
Bright One; Sun Ray; Shining; The Ancestor of the Hellenes; A Son of Deucalion and Pyrrha
Boy/Male
Gujarati, Hindu, Indian, Parsi
Lord Krishna
Girl/Female
Hindu, Indian
A Gopi
Male
Greek
(á¼Î²ÎµÎ») Greek form of Hebrew Hebel ("breath, breathing"), HABEL means "vanity," i.e. "transitory." In the bible, this is the name of the second son of Adam and Eve who was killed by his jealous brother Cain.
Boy/Male
American, Arabic, Australian, British, Chinese, Christian, English, Muslim
Happy; Battle; Short Form of the Biblical Shadrach
Boy/Male
Tamil
Lord Krishna, Moon
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
a.
Even; leaving no balance; as, to make or leave the accounts square.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
n.
To multiply by itself; as, to square a number or a quantity.
n.
Having the toe square.
n.
Hence, anything which is square, or nearly so
n.
A square; a measure; a rule.
n.
A square. See 1st Squire.
n.
A square piece or fragment.
a.
Having four equal sides and four right angles; as, a square figure.
n.
To make even, so as leave no remainder of difference; to balance; as, to square accounts.
imp. & p. p.
of Squire
imp. & p. p.
of Square
n.
The product of a number or quantity multiplied by itself; thus, 64 is the square of 8, for 8 / 8 = 64; the square of a + b is a2 + 2ab + b2.
v. t.
To attend as a squire.
n.
One who, or that which, squares.
n.
To place at right angles with the keel; as, to square the yards.
n.
To form with right angles and straight lines, or flat surfaces; as, to square mason's work.
n.
An instrument having at least one right angle and two or more straight edges, used to lay out or test square work. It is of several forms, as the T square, the carpenter's square, the try-square., etc.
a.
Forming a right angle; as, a square corner.
a.
Rendering equal justice; exact; fair; honest, as square dealing.