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Let $p_{j}$ denote the $j$-th prime and let
$$H_{m}=\liminf_{j\to\infty}(p_{j+m}-p_j).$$
For an integer $n$, define

Mervyn Tong recently asked me this question.
Say that a (first order) structure is $k$-ary if every formula is equivalent to a boolean combination of $k$-ary formulas. Equivalently: if it eliminates quantifiers in a $k$-ary relational language. There are examples of reducts of $k$-ary structures...

Let $R$ be a commutative integral domain and let $K$ be its field of fractions. Are the following statements equivalent?
If $f$ and $g$ are primitive polynomials in $R[x]$, then their product $fg$ is also primitive.
A primitive polynomial $f$ in $R[x]$ is irreducible in $R[x]$ if and only if it is irreducible in $K[x]$....

I have the following problem that I would like to address. Let M be a manifold with faces, this is a manifold with a given decomposition of the boundary i.e. $(M,\partial^{0,1}_iM)$ such that the intersection $\partial_iM\cap \partial_jM$ are again faces...