Integer-valued polynomial

In mathematics, an integer-valued polynomial (also known as a numerical polynomial) ${\displaystyle P(t)}$ is a polynomial whose value ${\displaystyle P(n)}$ is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial

${\displaystyle P(t)={\frac {1}{2}}t^{2}+{\frac {1}{2}}t={\frac {1}{2}}t(t+1)}$

takes on integer values whenever t is an integer. That is because one of t and ${\displaystyle t+1}$ must be an even number. (The values this polynomial takes are the triangular numbers.)

Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.