All Questions

Let $N=\prod_{k=0}^{k=m}{ p_k }$ be a square-free odd integer where $p_k$ is a prime. If we are given any integer $g$ such that $0<g<N$, it is very easy to tell if $g < \frac{N}{2}$ or not. ...

Is there any set of odd primes $\{p_1, p_2,..., p_k\}$ and natural numbers $a_1,..., a_k$ such that the following equation satisfied: $${p_1^{2a_1+1}+1 \over p_1+1}\times ....\times {p_k^{2a_k+1}+1 \...

Let $M$ be a finitely generated abelian group. Assume we are given a presentation of $M$, that is \begin{equation*} M = \frac{\bigoplus_{i=1}^r \mathbf{Z}g_i}{\sum_{j=1}^s \mathbf{Z} r_j} \end{...

This is the opposite to my last question case. Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ unramified in at least one infinite place of $F$. Let $\mathcal{O}⊂R$ be an ...

Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ ramified in all infinite places of $F$. Let $\mathcal{O}\subset R$ be an order. By assumption on $R$ its group of units $\...