$$
|A_i| := \vert\Bbb{Z}/(p_1) \times \dots \times \Bbb{Z}/(p_{i-1})\times \{\pm 1\} \times \Bbb{Z}/(p_{i+1}) \dots \times \Bbb{Z}/(p_n)\vert = 2\prod_{i \neq j} p_j
$$
is the size of the set of all $z \in \Bbb{Z}$, modulo $p_n\# = p_1 \cdots p_n$ such that $z^2 = 1 \pmod {p_i}$ but $z^2 \neq 1 \...
So in the Newton-Raphson method to iteratively approximate a root of a real polynomial, we start with a crude approximation $x_0 \in \mathbb{R}$ for $f(x)=0$ where $f(x) \in \mathbb{R}[x]$. For the next iterate $x_1$, we put $x_1 = x_0 + \epsilon$, and we want to determine $\epsilon$ to get a bet...
theorem:
let $ f:\mathbb{A} \to \mathbb{R} $ where $ \mathbb{A} \subseteq \mathbb{R} $ and let $c$ be a cluster point of $\mathbb{A}$ ,then:
if for every sequence $(x_{n})$ in $\mathbb{A}$ that converges to c such that $x_{n} \not=c $ for all $ n \in \mathbb{N}$, the sequence $(f(x_{n}))$ conver...
