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Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.

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How to prove this polynomial always has integer values at all integers?

.$$ For $k\geq0$ the $B_k$ are even polynomials of degree $2k$ that take integer values on integers. One has $B_k(k)=1$ for $k\geq1$, but $B_0(0)=2$. Further $B_k(i)=0$ for $|i|<k$. …
Wilberd van der Kallen's user avatar