integervalued

Integervalued is an informal mathematical term used to describe objects that take only integer values on their domain. In practice, the adjective is most often applied to functions, polynomials, or sequences that map into the set of integers. The concept encompasses several related ideas, depending on the context.

Integervalued functions are maps f from a domain such as a set of numbers to the integers, meaning f(x) ∈ Z for every x in the domain. Common examples include f(n) = n, f(n) = n^2, and other arithmetic expressions that yield integers for all inputs in the intended domain. Sequences that produce integer terms, such as a_n ∈ Z for all n, are another instance of the idea.

A prominent mathematical specialization is the study of integer-valued polynomials. These are polynomials with rational coefficients that nevertheless take integer values at every integer input; the set Int(Z) forms a subring of Q[x]. A standard result is that Int(Z) is freely generated as a Z-module by the binomial polynomials binomial(x, k), so any integer-valued polynomial can be uniquely expressed as a finite Z-linear combination of these binomial terms.

Applications of integervalued objects appear in interpolation problems over integers, combinatorial identities, and number-theoretic constructions where outputs must lie in the integers.

See also: integer-valued polynomial, Int(Z), binomial coefficients, polynomial functions, integer-valued sequences.