query: If $a-b$ divides the difference $f(a)-f(b)$, must $f$ be an integer polynomial?
positive_passages.docid: 144445
positive_passages.text: Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$
negative_passages.docid: 24098
negative_passages.text: Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
negative_passages.docid: 8164
negative_passages.text: Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q}$?
negative_passages.docid: 156940
negative_passages.text: There does not exist a polynomial $p(x)$ with integer coefficients which gives a prime number $\forall x\in \mathbb{Z}$
negative_passages.docid: 137669
negative_passages.text: $P$ is a monic polynomial of degree $n$ , then which are correct?
negative_passages.docid: 97652
negative_passages.text: Can every real polynomial be factored up to quadratic factors?
negative_passages.docid: 21006
negative_passages.text: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?
negative_passages.docid: 14270
negative_passages.text: Polynomial divisibility is unchanged by coeff. field extension
negative_passages.docid: 139017
negative_passages.text: If a polynomial takes integer values at every integer, does it have integer coefficients?
negative_passages.docid: 69728
negative_passages.text: If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$ for $R$ a sub-ring of $S$.
negative_passages.docid: 80502
negative_passages.text: $p(x)$ divided by $x-c$ has remainder $p(c)$? [Polynomial Remainder Theorem]
negative_passages.docid: 48873
negative_passages.text: Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.
negative_passages.docid: 158729
negative_passages.text: Are the polynomial remainder and factor theorems equivalent?
negative_passages.docid: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 99835
negative_passages.text: Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
negative_passages.docid: 65901
negative_passages.text: Solution verification to complex analysis problem, prove $f$ is a polynomial.
negative_passages.docid: 120573
negative_passages.text: Does $a!b!$ always divide $(a+b)!$