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Polynomial with integer value for integer input
mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle
Integer-valued_polynomial
integer-valued. In computer programming, many functions return values of integer type due to simplicity of implementation. Integer-valued polynomial Semi-continuity
Integer-valued_function
Type of mathematical expression
multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x {\displaystyle
Polynomial
formulas Integer-valued polynomial Algebraic equation Factor theorem Polynomial remainder theorem See also Theory of equations below. Polynomial ring Greatest
List_of_polynomial_topics
Complex number that solves a monic polynomial with integer coefficients
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 (a polynomial
Algebraic_integer
Number of subsets of a given size
takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials. The integer-valued polynomial 3t(3t
Binomial_coefficient
Analytic number theory conjecture
criterion for a polynomial f ( x ) {\displaystyle f(x)} in one variable with integer coefficients to give infinitely many prime values in the sequence
Bunyakovsky_conjecture
Irreducible polynomial whose roots are nth roots of unity
{\displaystyle n} -th cyclotomic polynomial, for any positive integer n {\displaystyle n} , is the unique irreducible polynomial with integer coefficients that is
Cyclotomic_polynomial
About polynomials in several variables
was formulated by Ott-Heinrich Keller in 1939, for the case of polynomials with integer coefficients. Arno van den Essen claims that Keller only talked
Jacobian_conjecture
Mathematical invariant of a knot or link
bracket polynomial is a Laurent polynomial in the variable A {\displaystyle A} with integer coefficients. First, we define the auxiliary polynomial (also
Jones_polynomial
Computational method
factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of
Factorization_of_polynomials
Amount left over after computation
is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder
Remainder
Algebraic structure
number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics
Polynomial_ring
Number theory conjecture
integer-valued polynomials (such as 1 2 x 2 + 1 2 x + 1 {\displaystyle {\tfrac {1}{2}}x^{2}+{\tfrac {1}{2}}x+1} , which takes integer values for all integers
Schinzel's_hypothesis_H
Index of articles associated with the same name
chromatic polynomial The flow polynomial, a polynomial whose values at integer arguments give the number of nowhere-zero flows with integer flow amounts
Graph_polynomial
Mathematical function
elementary symmetric polynomials. There is one elementary symmetric polynomial of degree d in n variables for each positive integer d ≤ n, and it is formed
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Pair of polynomial sequences
The Chebyshev polynomials are two sequences of orthogonal polynomials related to the cosine and sine functions, notated as T n ( x ) {\displaystyle T_{n}(x)}
Chebyshev_polynomials
Knot invariant
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander
Alexander_polynomial
Concept in complexity theory
runs in pseudo-polynomial time if its running time is bounded from above by a polynomial function of the two variables: the numeric value of the input (the
Pseudo-polynomial_time
arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Rational number Unit fraction Irreducible
List_of_number_theory_topics
Number in {..., –2, –1, 0, 1, 2, ...}
factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Error-detecting code for detecting data changes
of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation
Cyclic_redundancy_check
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
Matrix whose entries are integers
integer coefficients. Since the eigenvalues of a matrix are the roots of this polynomial, the eigenvalues of an integer matrix are algebraic integers
Integer_matrix
In mathematics, with negligible exceptions
surely Cahen, Paul-Jean; Chabert, Jean-Luc (3 December 1996). Integer-Valued Polynomials. Mathematical Surveys and Monographs. Vol. 48. American Mathematical
Almost_all
Measure of algorithmic complexity
difference between strongly- and weakly-polynomial time is when the inputs to the algorithms consist of integers or rational numbers. It is particularly
Strongly-polynomial_time
Unsolved problem in computer science
a list of distinct integers AND the integers are all in S AND the integers sum to 0 THEN OUTPUT "yes" and HALT This is a polynomial-time algorithm accepting
P_versus_NP_problem
Greatest common divisor of polynomials
divisor of two integers. In the important case of univariate polynomials over a field, the polynomial GCD may be computed as for the integer GCD, with the
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Type of symmetric polynomials in mathematics
Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Schur polynomials are indexed by integer partitions
Schur_polynomial
Mathematical concept
non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order
Degree_of_a_polynomial
of Alexandria, who made initial studies of integer Diophantine equations. An important type of polynomial Diophantine equations takes the form: s a +
Polynomial Diophantine equation
Polynomial_Diophantine_equation
Method to solve optimization problems
integral), and this optimal value may be found efficiently since all polynomial-size linear programs can be solved in polynomial time. One common way of proving
Linear_programming
Polynomials in combinatorial mathematics
Bell polynomial is equal to the number of ways the integer n can be expressed as a summation of k positive integers. This is the same as the integer partition
Bell_polynomials
Type of algebraic integer
integer α > 1 {\displaystyle \alpha >1} whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value
Salem_number
Number divisible only by 1 and itself
primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic
Prime_number
Polynomial equation whose integer solutions are sought
mathematics, a Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine
Diophantine_equation
Product with an integer
because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6. Each
Multiple_(mathematics)
Polynomial equation, generally univariate
equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions. Algebraic geometry is
Algebraic_equation
Type of complex number
algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example
Algebraic_number
Theorem about prime numbers
extended the Green–Tao theorem to cover polynomial progressions. More precisely, given any integer-valued polynomials P 1 , … , P k {\displaystyle P_{1},\ldots
Green–Tao_theorem
Root of a quadratic polynomial with a unit leading coefficient
some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic
Quadratic_integer
Sequence of differential equation solutions
non-negative integer. Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely
Laguerre_polynomials
Canonical solutions of the general Legendre equation
and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is
Associated Legendre polynomials
Associated_Legendre_polynomials
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
System of complete and orthogonal polynomials
integer, the solution Pn(x) that is regular at x = 1 is also regular at x = −1, and the series for this solution terminates (i.e. it is a polynomial)
Legendre_polynomials
Decomposition of a number into a product
Can integer factorization be solved in polynomial time on a classical computer? More unsolved problems in computer science In mathematics, integer factorization
Integer_factorization
Branch of mathematics
can be raised to a positive integer power. A monomial is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials
Algebra
Complexity class used to classify decision problems
subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in NP
NP_(complexity)
About simultaneous modular congruences
division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the
Chinese_remainder_theorem
Natural number
odd integers from 1 to 25 1,677,922,740 : number of series-reduced planted trees with 36 nodes 1,755,206,648 : coefficient of a ménage hit polynomial 1
1,000,000,000
Polynomials used for interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a
Lagrange_polynomial
Mathematical subject
Ziegler extended the result to cover polynomial progressions. More precisely, given any integer-valued polynomials P1,..., Pk in one unknown m all with
Arithmetic_combinatorics
Expression for sums of powers
positive integers ∑ k = 1 n k p = 1 p + 2 p + 3 p + ⋯ + n p {\displaystyle \sum _{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots +n^{p}} as a polynomial in n {\displaystyle
Faulhaber's_formula
Mathematical procedure
to Discover Integer Relations" (May 14, 2020) Weisstein, Eric W. "PSLQ Algorithm". MathWorld. A Polynomial Time, Numerically Stable Integer Relation Algorithm
Integer_relation_algorithm
resulting polynomial in z is at most dj, then condition 1 is satisfied. Proximity is preserved by the value function: There exists an integer G ≥ 0 (which
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
absolute constant μ > 1 {\displaystyle \mu >1} such that every polynomial with integer coefficients P ( x ) ∈ Z [ x ] {\displaystyle P(x)\in \mathbb {Z}
Lehmer's_conjecture
Natural number
prime number. For all integers n from 1 to 42, the value of 311 * log2(n) is within ¼ of an integer. 312 = 23 × 3 × 13. It is a Idoneal number and a practical
300_(number)
Polynomial invariant under variable permutations
symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally
Symmetric_polynomial
In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively
Positive_polynomial
Geometry of the location of polynomial roots
coefficients of the polynomial. Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Function in algebraic graph theory
{\displaystyle P(G,x)} is a monic polynomial of degree exactly n, with integer coefficients. The chromatic polynomial includes at least as much information
Chromatic_polynomial
Concepts from linear algebra
root of the characteristic polynomial, that is, the largest integer k such that (λi − λ)k evenly divides that polynomial. Suppose a matrix A has order
Eigenvalues_and_eigenvectors
Extension of the factorial function
OEIS. The values presented here are truncated rather than rounded.) The complex-valued gamma function is undefined for non-positive integers, but in these
Gamma_function
Natural number
oblong, kite, rhombus, and square. Four is the highest degree general polynomial equation for which there is a solution in radicals. Four is the only square
4
Relationship between the rational roots of a polynomial and its extreme coefficients
solutions of a polynomial equation a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0} with integer coefficients
Rational_root_theorem
Computation modulo a fixed integer
integer k (compatibility with exponentiation) p(a) ≡ p(b) (mod m), for any polynomial p(x) with integer coefficients (compatibility with polynomial evaluation)
Modular_arithmetic
The study of integer points in convex polyhedra is motivated by questions such as "how many nonnegative integer-valued solutions does a system of linear
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
Mathematical expression with disputed status
with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions. In other contexts, particularly in mathematical
Zero_to_the_power_of_zero
Algorithm for public-key cryptography
1) that allows the determination of d from e. No polynomial-time method for factoring large integers on a classical computer has yet been found, but it
RSA_cryptosystem
In mathematics, a non-algebraic number
number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental
Transcendental_number
Number with an integer power equal to 1
the nth cyclotomic polynomial. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if p
Root_of_unity
Mapping arbitrary data to fixed-size values
division by a polynomial modulo 2 instead of an integer to map n bits to m bits. In this approach, M = 2m, and we postulate an mth-degree polynomial Z(x) = xm
Hash_function
Visualization of the prime numbers
spiral correspond to polynomials of the form f ( n ) = 4 n 2 + b n + c {\displaystyle f(n)=4n^{2}+bn+c} where b and c are integer constants. When b is
Ulam_spiral
Estimate of time taken for running an algorithm
strongly polynomial time and weakly polynomial time algorithms. These two concepts are only relevant if the inputs to the algorithms consist of integers. The
Time_complexity
Function of the coefficients of a polynomial that gives information on its roots
discriminant is a polynomial in a 0 , … , a n {\displaystyle a_{0},\ldots ,a_{n}} with integer coefficients. When the above polynomial is defined over a
Discriminant
products of power-sum polynomials: they are a generating set over the rationals, but not over the integers. The power sum symmetric polynomial of degree k in
Power sum symmetric polynomial
Power_sum_symmetric_polynomial
On solvability of Diophantine equations
general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether
Hilbert's_tenth_problem
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
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
Arithmetic in a field with a finite number of elements
monic irreducible polynomial of degree n having coefficients in the finite field GF(q), where q = pt for some prime p and positive integer t, is called a
Finite_field_arithmetic
Cryptographic algorithm created by Adi Shamir
values that the polynomial may take by virtue of how it was constructed: the polynomial must have coefficients that are integers, and the polynomial must
Shamir's_secret_sharing
Hungarian-American mathematician and educator (1887–1985)
..49P. doi:10.1073/pnas.36.1.49. PMC 1063130. PMID 16588947. Integer-valued polynomial Laguerre–Pólya class Multivariate Pólya distribution Pólya's characterization
George_Pólya
Concept in number theory
factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization
Aurifeuillean_factorization
Polynomial related to differential operators
may have poles whenever b(s + n) is zero for a non-negative integer n. If f(x) is a polynomial, not identically zero, then it has an inverse g that is a
Bernstein–Sato_polynomial
Form of interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through
Polynomial_interpolation
Inherent difficulty of computational problems
process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an
Computational complexity theory
Computational_complexity_theory
Prime number of the form 2^n – 1
cyclotomic polynomial. The simplest generalized Mersenne primes are prime numbers of the form f(2n), where f(x) is a low-degree polynomial with small integer coefficients
Mersenne_prime
Arithmetic operation
operation for polynomials in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for
Division_(mathematics)
group of the ring of integers of a number field to the field's Dedekind zeta function. Casas-Alvero conjecture: if a polynomial of degree d {\displaystyle
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Algebraic structure with addition and multiplication
elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions,
Ring_(mathematics)
Number representing a continuous quantity
as the integer −5 and the fraction 4 / 3. Real numbers that are not rational are irrational. Those real numbers that are roots of polynomials with rational
Real_number
Association of one output to each input
multi-valued function of y that has three values for −2 < y < 2, and only one value for y ≤ −2 and y ≥ −2. Usefulness of the concept of multi-valued functions
Function_(mathematics)
(Mathematical) decomposition into a product
of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x2 − 4. Factorization is not
Factorization
Dutch mathematician and computer scientist
supervision of Hendrik Lenstra Jr. for the thesis, entitled "Integer Valued Polynomials in Algebraic Number Theory." After graduation, Zantema spent a
Hans_Zantema
List of mathematical contexts in which exponentiated terms are summed
terms of a Bernoulli polynomial. The triangular numbers are the sums of the first powers of the first several positive integers. The squares of the triangular
Sums_of_powers
Certain family of polynomials
are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that
Kostka_polynomial
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
Equation for the real part of a root of unity
every n, the polynomial Ψ n ( x ) {\displaystyle \Psi _{n}(x)} is monic, has integer coefficients, and is irreducible over the integers and the rational
Minimal polynomial of 2cos(2pi/n)
Minimal_polynomial_of_2cos(2pi/n)
Rational number sequence
Integer Sequences. 3 (1). Retrieved 11 February 2026. Kawasaki, Naho; Ohno, Yasuo (12 June 2023). "The Triangle Algorithm for Bernoulli Polynomials"
Bernoulli_number
Approximation of a function by a polynomial
by a polynomial of degree k {\textstyle k} , called the k {\textstyle k} -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the
Taylor's_theorem
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL
INTEGER VALUED-POLYNOMIAL