Let $f(x)$ be a polynomial with integer coefficients. Assume that $3$ divides the value $f(n)$ for each integer $n.$ Prove that when $f(x)$ is divided by $x^3-x, $ the remainder is of the form $3r(x),$ where $r(x)$ is a polynomial with integer coefficients. An Edited Solution is presented below f...

Could you give me advice on how to find a closed-form solution $t>0$ to the following transcendental equation: $$(t+1)^a - t^a = g$$ where $a>1$ and $g>1$. An accurate closed-form approximation for $t>0$ to see how $t$ depends on $a$ and $g$ could be also fine. I guess the above equation defines ...

Suppose $f$ is a analytic function defined on $\bar{D}(0;1)$ and has real value on the boundary. I'm trying to show $f$ can be extended to entire plane by $$g(z) = \begin{cases}f(z) &, \lvert z\rvert \leqslant 1\\ \frac{1}{\overline{f(\overline{z}^{-1})}}, &, \lvert z\rvert > 1\end{cases} $$ I t...

Schwarz's lemma: Suppose that $K\subset \mathbb C$ is an open connected set that is symmetric w.r.t. the real line. Let $K^+$ denote the part of $K$ above the real line and $K^-$; the part below the line. If $f_1: K^+\to \mathbb C$ and $f_2: K^-\to \mathbb C$ are holomorphic and extend continuo...