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Given an integer $n$, the Jacobsthal function $g(n)$ is the least integer, so that among any $g(n)$ consecutive integers $a,a+1,\dots,a+g(n)-1$ there is at least one that is coprime to $n$. Let $\nu(n …

At the cost of having the degree be very large you can always choose a $k$-gap with coefficients in $\lbrace 0,1\rbrace$. Pick a large $n$ so that $n\equiv -j\pmod{p_j}$, for all $1\le j\le k$. Where …

Here is a quick generating function argument. Start with the following lemma Lemma: The power series $$\frac{1}{(1-x)(1-2x)\cdots (1-kx)}$$ is even $\pmod{k+1}$ when $k+1$ is odd, and even $\pmo …

In both cases you are really only using the additive structure of your rings, so this is really a question about abelian groups. Assuming $n = p_1^{a_1} \cdots p_r^{a_r}$, when studying $A_n$ we are w …