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See searches and references containing INTEGER SQUARE-ROOT!INTEGER SQUARE-ROOT
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
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
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
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
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
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
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
(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
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 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
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
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
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
Determines the points needed for rasterizing a circle
{\displaystyle d_{n}^{2}} and the squared radius r 2 {\displaystyle r^{2}} to avoid having to compute an expensive and non-integer square-root. This also slightly changes
Midpoint_circle_algorithm
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
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
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
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
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
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
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
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
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
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
Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number
List_of_number_theory_topics
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)
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
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
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
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
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
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
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
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
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
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
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
Methods for locating real roots of a polynomial
polynomials with integer coefficients, and intervals ending with rational numbers. Also, the polynomials are always supposed to be square free. There are
Real-root_isolation
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)
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
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
Used to count, measure, and label
are integers (now called Gaussian integers) or rational numbers. His student, Gotthold Eisenstein, studied the type a + bω, where ω is a complex root of
Number
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
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A122400 (Number of square (0,1)-matrices without zero
1,000,000,000,000
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
147 107,890,609 = Wedderburn-Etherington number 111,111,111 = repunit, square root of 12345678987654321 111,111,113 = Chen prime, Sophie Germain prime,
100,000,000
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
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)
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)
Mathematical proof technique using contradiction
discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it. The square root of two is occasionally
Proof_by_infinite_descent
Integer side lengths of a right triangle
triple because the square root of 2 is not an integer. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple
Pythagorean_triple
Replacing a number with a simpler value
value of a function with integer domain and range. For example, if an integer n is known to be a perfect square, its square root can be computed by converting
Rounding
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
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
× 4 matrices with nonnegative integer entries and row and column sums equal to 3 2009 = 282 + 352, sum of two squares 2010 – number of compositions of
2000_(number)
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
Mathematical formula
the multiplicative digital root of n {\displaystyle n} . The multiplicative digital root for the first few positive integers are: 0, 1, 2, 3, 4, 5, 6,
Multiplicative_digital_root
quadratic Frobenius test (EQFT). Let n be a positive integer such that n is odd, and let b and c be integers such that ( b 2 + 4 c n ) = − 1 {\displaystyle
Quadratic_Frobenius_test
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
Family of related bitwise operations on machine words
Warren. Chapter 11-1: Integer Square Root. Schlegel, Benjamin; Gemulla, Rainer; Lehner, Wolfgang [in German] (June 2010). "Fast integer compression using
Find_first_set
algorithm De Casteljau's algorithm Square roots and other roots: Integer square root Methods of computing square roots nth root algorithm hypot — the function
List of numerical analysis topics
List_of_numerical_analysis_topics
the square root of 8 can be represented with a simple repeating pattern of integers: Because the square root of nine is a rational number, the square root
Square_root_of_8
Number raised to the third power
cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64. Unlike perfect squares, perfect cubes do not have a small
Cube_(algebra)
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
(Mathematical) decomposition into a product
this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra
Factorization
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)
Natural number between 89 and 91
(6^{2}+5^{2}+4^{2}+3^{2}+2^{2})} . The square of eleven 112 = 121 is the ninetieth indexed composite number, where the sum of integers { 2 , 3 , . . . , 11 } {\displaystyle
90_(number)
1617 device for calculating products and quotients
less than 11669900, so the root needs to be rounded up to 6840.0. To find the square root of a number that isn't an integer, say 54782.917, everything
Napier's_bones
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
Function in mathematical number theory
\varphi } -root modulo n.) Theorem 2—For every positive integer n there exists a primitive λ-root modulo n. Moreover, if g is such a root, then there
Carmichael_function
Square of numbers with equal row, column and diagonal totals
historical and recreational mathematics, a magic square is a square array of numbers, usually positive integers, where the sums of the numbers in each row,
Magic_square
Natural number
the square of 24. While 17 different integers have a totient value of 72, the sum of Euler's totient function φ(x) over the first 15 integers is 72
72_(number)
Function of the coefficients of a polynomial that gives information on its roots
square root in the quadratic formula. If a ≠ 0 , {\displaystyle a\neq 0,} this discriminant is zero if and only if the polynomial has a double root.
Discriminant
Multiplication table in Indian mathematics
mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the
Vedic_square
Repeated sum of a number's digits
b; x /= b; } return total; } // Digital root in base b static int digitalRoot(int x, int b) { HashSet<Integer> seen = new HashSet<>(); while (!seen.contains(x))
Digital_root
Integer
concept of abstract algebra generalizing integers and real numbers. Although there are no real number square roots of −1, the complex number i satisfies
−1
Mathematical result of division
not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square. Outside of arithmetic
Quotient
Irrational numbers which appear to be rational
ratio of two integers. Transcendental numbers like e and π, and noninteger surds such as square root of 2 are irrational.) Almost integer Normal number
Schizophrenic_number
Result in modular arithmetic
infinity, it follows that a root or a factorization modulo p can be lifted to a root or a factorization over the p-adic integers. These results have been
Hensel's_lemma
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
Irreducible polynomial whose roots are nth roots of unity
polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( e 2 i
Cyclotomic_polynomial
Natural number
number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "Curium | Cm (Element)
10,000,000
Conjecture in number theory
conjecture on primitive roots states that a given integer a that is neither a square number nor −1 is a primitive root modulo infinitely many primes p. The conjecture
Artin's conjecture on primitive roots
Artin's_conjecture_on_primitive_roots
Number whose cube is a given number
real cube roots of 8 and −8 are respectively 2 and −2. The real cube root of an integer or of a rational number is generally not a rational number, nor a
Cube_root
depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial P irreducible
Totally_real_number_field
Counting polynomial roots in an interval
polynomial of odd degree. In the case of a non-square-free polynomial, if neither a nor b is a multiple root of p, then V(a) − V(b) is the number of distinct
Sturm's_theorem
Lemma concerning the limit of subadditive sequences
In mathematics, and in particular in calculus, Fekete's lemma (also called Fekete's subadditive lemma) is a lemma concerning the limit of subadditive sequences
Fekete's_lemma
Measures the size of the ring of integers of the algebraic number field
measures the size of the (ring of integers of the) algebraic number field. More specifically, it is proportional to the squared volume of the fundamental domain
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Geometric shape
the square root of an integer, such as √2, √3, etc. The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a square to
Dynamic_rectangle
Natural number
digits of the square of 49 (2401) is the square root of 49. 49 is the first square where the digits are squares. In this case, 4 and 9 are squares. The fraction
49_(number)
Natural number
primes, 161 is a Blum integer. 161 is a palindromic number. 161/72 is a commonly used rational approximation of the square root of 5 and is the closest
161_(number)
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Boy/Male
French Latin
A squire.
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Boy/Male
Arabic, Muslim
To Wait
Boy/Male
English American
Shieldbearer.
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
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.
Surname or Lastname
English
English : variant of Squire.
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Boy/Male
British, English
Spear-man
Surname or Lastname
English
English : variant of Spear.
Boy/Male
American, British, English
Shield Bearer
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Boy/Male
Italian
Squire.
Boy/Male
Anglo Saxon American English Scottish
Steward.
Surname or Lastname
English
English : patronymic from Squire.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Boy/Male
English
Shieldbearer.
Boy/Male
Muslim
To wait
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
Biblical
Breaking; bruising small; gold; coloring
Boy/Male
Indian, Telugu
Love
Boy/Male
Arthurian Legend
Name of a king.
Boy/Male
African, American, Australian, British, Chinese, Christian, English, German, Greek, Irish, Latin
Merciful; Inventor of the Corn Mill; Servant; Soldier
Boy/Male
Hindu, Indian, Tamil, Telugu
Special Knowledge; Pride
Boy/Male
Indian, Punjabi, Sikh
The Word of God
Surname or Lastname
English
English : from a pet form of the female personal name Barbara (see Barbara).Southern French : from a diminutive of Occitan barbari ‘barbarous’, ‘barbarian’. In particular, this word came to denote a Moor or Berber from the Barbary Coast in North Africa, and hence was then applied to a man of swarthy appearance or uncouth habits.An immigrant from the Périgord region of France was variously documented in Montreal in 1668 as Barbary and Barbarin, with the secondary surname Grandmaison.
Boy/Male
Muslim
Tall
Girl/Female
Gujarati, Hindu, Indian, Japanese, Kannada, Malayalam, Marathi, Tamil, Telugu
Fame; Righteous; Good Faith; Victorious; Famous; Successful
Boy/Male
Hindu, Indian
God Indran; Jeya means Victory; Indran is God
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
INTEGER SQUARE-ROOT
a.
Having four equal sides and four right angles; as, a square figure.
n.
To multiply by itself; as, to square a number or a quantity.
a.
Even; leaving no balance; as, to make or leave the accounts square.
n.
A square piece or fragment.
imp. & p. p.
of Square
n.
To form with right angles and straight lines, or flat surfaces; as, to square mason's work.
a.
Forming a right angle; as, a square corner.
n.
To make even, so as leave no remainder of difference; to balance; as, to square accounts.
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.
a.
Rendering equal justice; exact; fair; honest, as square dealing.
n.
A square; a measure; a rule.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
v. t.
To attend as a squire.
n.
Hence, anything which is square, or nearly so
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.
n.
One who, or that which, squares.
n.
To place at right angles with the keel; as, to square the yards.
n.
A square. See 1st Squire.
n.
Having the toe square.
imp. & p. p.
of Squire