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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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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
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
test called extended 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
Quadratic_Frobenius_test
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
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
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
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
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)
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A006203 (Discriminants of imaginary quadratic fields with class
307_(number)
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
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
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
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
Optimization solver
(LP), quadratic programming (QP), quadratically constrained programming (QCP), mixed integer linear programming (MILP), mixed-integer quadratic programming
Gurobi_Optimizer
Quadratic imaginary number fields with unique factorisation
theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special
Stark–Heegner_theorem
Branch of pure mathematics
rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions
Number_theory
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
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)
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
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)
Subfield of mathematical optimization
all linear. Quadratic programming are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function. Second
Convex_optimization
Principal square root of minus 1
usually denoted by i, is a mathematical constant that is a solution to the quadratic equation x2 = −1, which is not solved by any real number. Any real-number
Imaginary_unit
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
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
Felix Klein. The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field Q ( − 7 ) {\displaystyle
Kleinian_integer
In algebra, element without non-trivial factors
a prime (hence irreducible) ideal of D {\displaystyle D} . In the quadratic integer ring Z [ − 5 ] , {\displaystyle \mathbf {Z} [{\sqrt {-5}}],} it can
Irreducible_element
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
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
(Mathematical) decomposition into a product
smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of
Factorization
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
Symbol in number theory
0'=1} ). Then the following symmetric version of quadratic reciprocity holds for every pair of integers m , n {\displaystyle m,n} such that gcd ( m , n
Kronecker_symbol
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
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)
Type of group in mathematics
{T}}=I\right\}.} More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form
Orthogonal_group
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
Natural number
a number, numeral, and grapheme. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property
1
Unsolved problem in mathematics
algorithm to decide whether a system of diagonal quadratic forms with integer coefficients represents an integer tuple. Indeed, Büchi observed that squaring
Büchi's_problem
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)
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
Male
German
German surname transferred to forename use, derived from the personal name Liutbert, LUBBERT means "people-bright."
Girl/Female
Biblical
Dart of joy, division of a song.
Boy/Male
Tamil
Lord of the whole world, Lord Ganesh
Female
Italian
Italian form of Greek Elisabet, ELISABETTA means "God is my oath."
Boy/Male
Muslim
Clear
Boy/Male
Welsh
Sacrifice.
Girl/Female
Hindu, Indian
Goddess Saraswati
Boy/Male
Hindu
Arnik
Girl/Female
Bengali, Gujarati, Hindu, Indian, Telugu
Beautiful
Boy/Male
Hindu, Indian, Vietnamese
Good; Perfect
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
QUADRATIC INTEGER
pl.
of Quadratrix
a.
Tetragonal.
n.
A curve made use of in the quadrature of other curves; as the quadratrix, of Dinostratus, or of Tschirnhausen.
a.
Of or pertaining to a square, or to squares; resembling a quadrate, or square; square.
a.
The quadrate bone.
a.
Of or pertaining to the quadrate and jugal bones.
a.
To square; to agree; to suit; to correspond; -- followed by with.
p. pr. & vb. n.
of Quadrate
a.
Pertaining to terms of the second degree; as, a quadratic equation, in which the highest power of the unknown quantity is a square.
a.
Of or pertaining to the biquadrate, or fourth power.
n.
A biquadrate.
pl.
of Quadratrix
n.
That branch of algebra which treats of quadratic equations.
a.
A quadrate; a square.
n.
A quadrat.
a.
Quadrate; square.
imp. & p. p.
of Quadrate
n.
Same as Quadrate.
v. t.
To adjust (a gun) on its carriage; also, to train (a gun) for horizontal firing.
n.
A biquadratic equation.