Pearl Dive

Let $p$ be an odd prime, and let $n_1,\dots,n_{p-2}, m$ be even integers such that $n_1 < n_2 < \dots < n_{p-2}$ and \begin{equation} 2m > \sum_{i=1}^{p-2} n_i^2. \end{equation} Consider the polynomial \begin{equation} g(x)=(x^2 + m)(x - n_1) \dots (x - n_{p-2}). \end{equation} From Rolle's Theor...

Let $P$ and $Q$ be monic polynomials with integer coefficients and degrees $n$ and $d$ respectively, where $d\mid n$. Suppose there are infinitely many pairs of positive integers $(a,b)$ for which $P(a)=Q(b)$. I would like to determine if exists a polynomial $R$ with integer coefficients such t...

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to construct such a quartic and rule out the existence of such a quadratic, but this leaves open the quest...

I'm going by the maxim Groups, like men, are known by their actions This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does one visualize the action of $G \rtimes H$? What does it act on? Some combination of $S$ and $T...

Is there any integer solution for $$\operatorname{Re}((a+bi)^{m})=\operatorname{Re}((a+bi)^{n})$$ except $(m,n)=(1,3)$, where $0\leq m<n,\ |a|\neq |b|,\ a\neq 0,\ b\neq 0$? In other words, Can $\operatorname{Re}(a+bi)^{n}$ be overlapped with $a+bi\in\mathbb{Z}[i]$ fixed? This is a ge...

Real projective space $\mathbb{R}P^n$ embeds in a natural way in complex projective space $\mathbb{C}P^n$. (Using standard projective coordinates on $\mathbb{C}P^n$, $\mathbb{R}P^n$ is the subspace consisting of points that have a representative in which all coordinates are real.) I know (very ...

Suppose there is a grid $[1,N]^2$. A person standing at some initial point $(x_0,y_0)$ walk randomly within the grid. At each location, he/she walks to a neighboring location with equal probability (e.g., for an interior point, the probability is $\frac{1}{4}$; for a corner, it's $\frac{1}{2}$.)....

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my question is as in the title: Do any two rational triangles of the same area have a common side-rationa...

Motivated by this paper on polynomial continued fractions (Bowman, 2000), I thought about various extensions to current definitions of these fractions. What if we defined a continued fraction such that the numerator of each fraction is the integral of the previous one? That is, define th...