Integer-valued polynomial

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In mathematics, an integer-valued polynomial P(t) is a polynomial taking an integer value P(n) for every integer n. Certainly every polynomial with integer coefficients is integer-valued. There are simple examples to show that the converse is not true: for example the polynomial

t(t + 1)/2

giving the triangle numbers takes on integer values whenever t = n is an integer. That is because one out of n and n + 1 must be an even number.

In fact integer-valued polynomials can be described fully. Inside the polynomial ring Q[t] of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials

P_k(t) = t(t − 1)...(t − k + 1)/k!

for k = 0,1,2, ... .

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