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### When are the integers a free R-module?

If $\mathbb{Z}$ is a free $R$-module, then as a group, $\mathbb{Z} \cong \bigoplus_n R$. $\mathbb{Z}$ isn't a direct sum, except trivially, so $\mathbb{Z} \cong R$ as groups. So addition on $R$ has ...

### Do large sets have this specific type of self-similarity?

Yes, such self-similarity must exist. In fact, we can even take $d = 1$. Let $A$ denote this sequence. The point is that, if the desired self-similarity does not exist for some $k$, then for each ...
1 vote

### Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$

Computing $(x+x^2+x^3+x^4+x^5+x^6)^{10}$ with Wolfram gives a coefficient of $2930455$ to $x^{30}$.
1 vote

### Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$

Following my comment, you can use Principle of Inclusion Exclusion. First find all ways (no upper bound on $x_i$), then subtract off number of ways given $x_1 > 5,$ given $x_2 > 5, \dots,$ given ...

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