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In mathematics, an integer-valued polynomial (also known as a numerical polynomial) ''P''(''t'') is a polynomial whose value ''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 : takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and ''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.〔. See in particular pp. 213–214.〕 ==Classification== The class of integer-valued polynomials was described fully by . Inside the polynomial ring ''Q''() of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials :''Pk''(''t'') = ''t''(''t'' − 1)...(''t'' − ''k'' + 1)/''k''! for ''k'' = 0,1,2, ..., i.e., the binomial coefficients. In other words, every integer-valued polynomial can be written as an integer linear combination of binomial coefficients in exactly one way. The proof is by the method of discrete Taylor series: binomial coefficients are integer-valued polynomials, and conversely, the discrete difference of an integer series is an integer series, so the discrete Taylor series of an integer series generated by a polynomial has integer coefficients (and is a finite series). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Integer-valued polynomial」の詳細全文を読む スポンサード リンク
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