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Root of a quadratic polynomial with a unit leading coefficient
number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a
Quadratic_integer
Field (mathematics) generated by the square root of an integer
square-free integer different from 0 {\displaystyle 0} and 1 {\displaystyle 1} . If d > 0 {\displaystyle d>0} , the corresponding quadratic field is called
Quadratic_field
Mathematical concept
quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers
Quadratic_irrational_number
Complex number whose mapping on a coordinate plane produces a triangular lattice
rounding-to-integer functions. The reason this satisfies N(ρ) < N(β), while the analogous procedure fails for most other quadratic integer rings, is as
Eisenstein_integer
Number with an integer power equal to 1
unity) is a quadratic integer. For n = 5, 10, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum
Root_of_unity
Integer that is a perfect square modulo some integer
theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that
Quadratic_residue
Algorithm for computing greatest common divisors
ideals. The quadratic integer rings are helpful to illustrate Euclidean domains. Quadratic integers are generalizations of the Gaussian integers in which
Euclidean_algorithm
Complex number whose real and imaginary parts are both integers
Gaussian integers do not have a total order that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers
Gaussian_integer
Polynomial with all terms of degree two
quadratic form on a vector space. The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic
Quadratic_form
Complex number that solves a monic polynomial with integer coefficients
{\frac {1}{2}}(1+{\sqrt {d}}\,)} respectively. See Quadratic integer for more. The ring of integers of the field F = Q [ α ] {\displaystyle F=\mathbb {Q}
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
Number, approximately 1.618
of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, z + z ¯ {\displaystyle z+{\bar {z}}} , is a quadratic integer
Golden_ratio
Integer factorization algorithm
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field
Quadratic_sieve
Solving an optimization problem with a quadratic objective function
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks
Quadratic_programming
Type of complex number
algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified as quadratic integers. Gaussian integers, complex numbers a
Algebraic_number
Condition under which an odd prime is a sum of two squares
rings of quadratic integers. In summary, if O d {\displaystyle {\mathcal {O}}_{\sqrt {d}}} is the ring of algebraic integers in the quadratic field, then
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Quadratic homogeneous polynomial in two variables
in quadratic form. A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form
Binary_quadratic_form
Algebraic construction
{\displaystyle d} is a square-free integer and K = Q ( d ) {\displaystyle K=\mathbb {Q} ({\sqrt {d}}\,)} is the corresponding quadratic field, then O K {\displaystyle
Ring_of_integers
Number whose square is a given number
major use in the formula for solutions of a quadratic equation. Quadratic fields and rings of quadratic integers, which are based on square roots, are important
Square_root
Natural number
} 33 is the last of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}
33_(number)
Natural number
Ramanujan–Nagell equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}
7
Computation modulo a fixed integer
totient function. Quadratic residue: An integer a is a quadratic residue modulo m, if there exists an integer x such that x2 ≡ a (mod m). Euler's criterion
Modular_arithmetic
Method to solve optimization problems
Abstraction of ordered linear algebra Quadratic programming – Solving an optimization problem with a quadratic objective function Semidefinite programming –
Linear_programming
Optimization problem in mathematics
Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since
Quadratically constrained quadratic program
Quadratically_constrained_quadratic_program
Mathematical proportionality to a square
real-valued function of an integer or natural number variable). Examples of quadratic growth include: Any quadratic polynomial. Certain integer sequences such as
Quadratic_growth
Topics referred to by the same term
martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern integer factorization
Quadratic
Polynomial equation of degree two
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle
Quadratic_equation
In mathematics, element with a multiplicative inverse
constitute the multiplicative group of integers modulo n. In the ring Z[√3] obtained by adjoining the quadratic integer √3 to Z, one has (2 + √3)(2 − √3) =
Unit_(ring_theory)
Nearest integers from a number
returns the greatest integer less than or equal to x, written ⌊x⌋ or floor(x). Similarly, the ceiling function returns the least integer greater than or equal
Floor_and_ceiling_functions
Sum type in number theory
and applied them to quadratic, cubic, and biquadratic reciprocity laws. For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is
Quadratic_Gauss_sum
Gives conditions for the solvability of quadratic equations modulo prime numbers
symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a ( mod p ) {\displaystyle x^{2}\equiv
Quadratic_reciprocity
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
Positive real number which when multiplied by itself gives 5
-{\sqrt {5}}} , it solves the quadratic equation x 2 − 5 = 0 {\displaystyle x^{2}-5=0} , making it a quadratic integer, a type of algebraic number.
Square_root_of_5
Mathematical structure in abstract algebra
that square root. A quadratic integer ring (for some D) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras
*-algebra
On when an integer positive definite quadratic form represents all positive integers
definite quadratic form arising from an integer matrix represents all positive integers up to 15, then it represents all positive integers. Conway and
15_and_290_theorems
Function of the coefficients of a polynomial that gives information on its roots
a discriminant is equivalent to a unique square-free integer. By a theorem of Jacobi, a quadratic form over a field of characteristic different from 2
Discriminant
Natural number
\mathbb {F_{1}} } ), and 73 the largest indexed member of a definite quadratic integer matrix representative of all prime numbers that is also the number
72_(number)
Product of two distinct primes ≡ 3 (mod 4)
a number with 65,886,368 digits. 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
Blum_integer
Formula concerning prime numbers
for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then a p −
Euler's_criterion
Integers have unique prime factorizations
factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely as a product
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Number system extending the rational numbers
square root of an integer that is a quadratic residue modulo p. This seems to be the fastest known method for testing whether a large integer is a square:
P-adic_number
Natural number
290 theorems describes integer-quadratic matrices that describe all positive integers, by the set of the first fifteen integers, or equivalently, the first
29_(number)
Xpress – solver for linear and quadratic programming with continuous or integer variables (MIP). FortMP – linear and quadratic programming. FortSP – stochastic
List_of_optimization_software
Natural number
{\displaystyle 7} ; this sum represents the largest square-free integer over a quadratic field of class number two, where 163 is the largest such (Heegner)
21_(number)
Natural number
nine Heegner numbers, or square-free positive integers n {\displaystyle n} that yield an imaginary quadratic field Q [ − n ] {\displaystyle \mathbb {Q} \left[{\sqrt
9
In number theory, measure of non-unique factorization
binary quadratic forms is isomorphic to the narrow class group of Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})} . For real quadratic integer rings
Ideal_class_group
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
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
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Subfield of convex optimization
crossing from one side to the other. This problem can be expressed as an integer quadratic program: Maximize ∑ ( i , j ) ∈ E 1 − v i v j 2 , {\displaystyle \sum
Semidefinite_programming
Natural number
of Integer Sequences. OEIS Foundation. Retrieved 26 December 2022. Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields
23_(number)
Mathematical proof technique
two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine
Vieta_jumping
Finite extension of the rationals
\mathbb {Q} } . More generally, for any square-free integer d {\displaystyle d} , the quadratic field Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})}
Algebraic_number_field
Number divisible only by 1 and itself
concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a
Prime_number
Type of algebraic integer
simply PV number. For example, the golden ratio, φ ≈ 1.618, is a real quadratic integer that is greater than 1, while the absolute value of its conjugate
Pisot–Vijayaraghavan_number
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
Commutative ring with no zero divisors other than zero
irreducible. The converse is not true in general: for example, in the quadratic integer ring Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]}
Integral_domain
Natural number
positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290
15_(number)
Unique algebraic expression given by Srinivasa Ramanujan
concepts and arguments. Every integer of the form 10n + 5 is represented by Ramanujan's ternary quadratic form. If n is an odd integer which is not square-free
Ramanujan's ternary quadratic form
Ramanujan's_ternary_quadratic_form
Function in number theory
odd prime number and a {\displaystyle a} is a positive integer that may or may not be a quadratic residue mod p. The Legendre symbol is a multiplicative
Legendre_symbol
Method for computing the relation of two integers with their greatest common divisor
When using integers of unbounded size, the time needed for multiplication and division grows quadratically with the size of the integers. This implies
Extended_Euclidean_algorithm
Study of mathematical algorithms for optimization problems
variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set. A problem with
Mathematical_optimization
Integer side lengths of a right triangle
of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of x2 + y2 − z2 over the integers. Alternatively
Pythagorean_triple
algebraic numbers, and include the quadratic surds. Algebraic integer: A root of a monic polynomial with integer coefficients. Transfinite numbers: Numbers
List_of_types_of_numbers
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective
Quadratic_knapsack_problem
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
Rational numbers with root 5 added
{\displaystyle \textstyle \varphi ^{n}} for any non-zero integer n {\displaystyle n} . The quadratic polynomial x 2 − 5 F n x + ( − 1 ) n + 1 5 {\displaystyle
Golden_field
Product of a number by itself
place of x2. The adjective which corresponds to squaring is quadratic. The square of an integer may also be called a square number or a perfect square. In
Square_(algebra)
Type of integral domain
w, the ring R[X1, ..., Xn, Z]/(Zc − F(X1, ..., Xn)) is a UFD. The quadratic integer ring Z [ − 5 ] {\displaystyle \mathbb {Z} [{\sqrt {-5}}]} of all complex
Unique_factorization_domain
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
Problem in computational number theory
The quadratic residuosity problem (QRP) in computational number theory is to decide, given integers a {\displaystyle a} and N {\displaystyle N} , whether
Quadratic_residuosity_problem
Prime number congruent to 1 mod 4
then p {\displaystyle p} is a quadratic residue mod q {\displaystyle q} if and only if q {\displaystyle q} is a quadratic residue mod p {\displaystyle
Pythagorean_prime
Algebraic structure with addition and multiplication
is commutative. The ring of quadratic integers, the integral closure of Z {\displaystyle \mathbb {Z} } in a quadratic extension of Q . {\displaystyle
Ring_(mathematics)
Locus of the zeros of a polynomial of degree two
quadrics have dimension two, and are known as quadric surfaces. Their quadratic equations have the form m x x x 2 + m y y y 2 + m z z z 2 + 2 m x y x
Quadric
Number that is not a ratio of integers
irrational numbers are those that cannot be expressed as the ratio of two integers. Geometrically, when the ratio of lengths of two line segments is an irrational
Irrational_number
Prime number of the form 2^n – 1
of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied
Mersenne_prime
Algebraic structure in mathematics
ring Z of integers, then the quadratic algebra Z [ X ] / ( X 2 + 1 ) {\displaystyle \mathbb {Z} [X]/(X^{2}+1)} is called the Gaussian integers. If R is
Quadratic_algebra
Two quadratic forms over a number field are equivalent iff they are equivalent locally
local fields. Basic invariants of a nonsingular quadratic form are its dimension, which is a positive integer, and its discriminant modulo the squares in
Hasse–Minkowski_theorem
Generalization of golden and silver ratios
algebraic number theory, the metallic means are exactly the real quadratic integers that are greater than 1 {\displaystyle 1} and have − 1 {\displaystyle
Metallic_mean
Combinatorial optimization problem
terms of quadratic inequalities, hence the name. The formal definition of the quadratic assignment problem is as follows. Given a positive integer n {\displaystyle
Quadratic_assignment_problem
Estimate of time taken for running an algorithm
general-purpose sorts run in linear time, but the change from quadratic to sub-quadratic is of great practical importance. An algorithm is said to be of
Time_complexity
Number representing a continuous quantity
Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established the existence of transcendental numbers;
Real_number
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
Combinatorial optimization problem
Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem
Quadratic unconstrained binary optimization
Quadratic_unconstrained_binary_optimization
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A076409 (Sum of the quadratic residues of prime(n))". The
1000_(number)
Natural number
1729 can be expressed as the quadratic form. Investigating pairs of its distinct integer values that represent every integer the same number of times, Schiemann
1729_(number)
Number, approximately 2.41421
mutandis for all quadratic Pisot numbers that satisfy the general equation x 2 = n x + 1 , {\displaystyle x^{2}=nx+1,} with integer n > 0. It follows
Silver_ratio
Algorithm for finding zeros of functions
Furthermore, for a root of multiplicity 1, the convergence is at least quadratic (see Rate of convergence) in some sufficiently small neighbourhood of
Newton's_method
Natural number
Integer Sequences. OEIS Foundation. 2016-04-18. Retrieved 2016-04-19. Sloane, N. J. A. (ed.). "Sequence A046002 (Discriminants of imaginary quadratic
131_(number)
Relationship between the rational roots of a polynomial and its extreme coefficients
{a^{2}}{b}}} and b 2 a {\displaystyle {\tfrac {b^{2}}{a}}} must be integer. Consider the quadratic equation whose roots are a 2 b {\displaystyle {\tfrac {a^{2}}{b}}}
Rational_root_theorem
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields with class
307_(number)
Family of solutions to related differential equations
are when α {\displaystyle \alpha } is an integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often
Bessel_function
Factorization method based on the difference of two squares
named after 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}
Fermat's_factorization_method
Mathematical concept
genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently
Genus_of_a_quadratic_form
Address collision resolution scheme
Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking
Quadratic_probing
Negative integer two units from the origin in mathematics
the class number of the quadratic field Q [ d ] {\displaystyle \mathbb {Q} [{\sqrt {d}}]} equal to 1, meaning its ring of integers is a unique factorization
−2
Theorem in functional analysis
{\displaystyle \|A\|_{\infty \to 1}} is by solving the following quadratic integer program: max ∑ i , j A i j x i y j s.t. ( x , y ) ∈ { − 1 , 1 } m
Grothendieck_inequality
Listing all imaginary quadratic fields with a given class number
problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields Q ( d ) {\displaystyle
Class_number_problem
Visualization of the prime numbers formed by arranging the integers into a spiral
American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers. Ulam and Gardner
Ulam_spiral
17th-century conjecture proved by Andrew Wiles in 1994
conjecture, especially in older texts) states that there are no positive integers a , b , c , n {\displaystyle a,b,c,n} with n > 2 {\displaystyle n>2} such
Fermat's_Last_Theorem
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
Boy/Male
Indian, Punjabi, Sikh
The Highest God
Boy/Male
Australian, Danish, Dutch, Finnish, French, German, Greek, Latin, Swedish
Laurentium was a City South of Rome Known for Its Numerous Laurel Trees; Man from Laurentum
Boy/Male
Hindu
Always
Boy/Male
English
Craftsman.
Girl/Female
Latin American
Happy. Feminine of Felix.
Girl/Female
American, Arabic, Australian, British, Chinese, Danish, English, Finnish, French, German, Greek, Indian, Muslim, Swedish, Tamil
Will; Desire; Helmet; God's Protection; Will-helmet; Apple; Protection
Surname or Lastname
English
English : possibly a habitational name from a lost or unidentified place, most likely in Dorset or Somerset, where the surname occurs most frequently. Alternatively, it may be from the Old English personal name CynestÄn.
Girl/Female
Arabic, French
Giving Counsel; Advising
Boy/Male
Australian, Danish, Swedish
Appointed
Girl/Female
Indian
Gold
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
a.
The quadrate bone.
a.
Quadrate; square.
p. pr. & vb. n.
of Quadrate
n.
A quadrat.
pl.
of Quadratrix
a.
Of or pertaining to the biquadrate, or fourth power.
n.
That branch of algebra which treats of quadratic equations.
n.
A biquadratic equation.
imp. & p. p.
of Quadrate
a.
To square; to agree; to suit; to correspond; -- followed by with.
a.
Pertaining to terms of the second degree; as, a quadratic equation, in which the highest power of the unknown quantity is a square.
n.
A biquadrate.
a.
Tetragonal.
pl.
of Quadratrix
n.
Same as Quadrate.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
n.
A curve made use of in the quadrature of other curves; as the quadratrix, of Dinostratus, or of Tschirnhausen.
a.
A quadrate; a square.
a.
Of or pertaining to the quadrate and jugal bones.
v. t.
To adjust (a gun) on its carriage; also, to train (a gun) for horizontal firing.