Questions tagged [modular-arithmetic]

Consider $\mathbb{Z}_q \equiv \mathbb{Z}/q\mathbb{Z}$, where $q \geqslant 2$. A set of vectors in $\mathbb{Z}_q^n$ is said to be linearly independent if no nontrivial linear combination of them ...

$\DeclareMathOperator\len{len}$Let $a, b \in \mathbb{N} -\{ 0, 1 \}$ and define ${^{b}a}$ to be $a^a$ if $b = 2$ and $a^{\left(^{b-1}a \right)}$ if $b \geq 3$ (e.g., ${^{3}5} = 5^{\left( 5^5 \right)} =...

Given a monic irreducible polynomial $f\in\mathbb{Z}[x]$, I'd like to know for how many primes p we have that $f \bmod p$ is irreducible. In the link: How many primes stay inert in a finite (non-...

A powerful number is an integer $m$ such that if $p$ is prime and $p \mid m$ then $p^2 \mid m$. Powerful numbers can be represented in the form $m=u^2 v^3$. Erdos conjectured that three consecutive ...

Fix a positive integer $r$. Describe the solutions to the system of equations given by: $$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$ Example: In the case ...

This is a question I also asked on MSE . If it is frowned upon to ask the same question on both threads, you can vote to close this thread. Let $n \in \mathbb {N^*}$ and $$S_n = \sum_{k=1}^{n-1} (k^2 \...