Pearl Dive

Every one of the functions $\sin(x), \ldots, \sin(x+n)$ solves the ODE $y'' + y = 0$. But the solution space of a linear (homogeneous) second order ODE is only two-dimensional, which tells you that these sine functions must be linearly dependent if you take at least $3$ of them. That is, there ex...

Curators Much of the effort towards improving question quality goes towards sanctioning bad questions-- closure, review, etc. There is a lot of room for greater recognition of good questions. On our site, we have experimented with ways to do this in the form of curators. Curators seek out excepti...

Let $H$ and $N$ be two groups, $\phi$ and $\psi$ be two homomorphisms $H \to \operatorname{Aut}(N)$, and $G_1 = N \rtimes_\phi H$ and $G_2 = N \rtimes_\psi H$ corresponding semidirect products. What are conditions on $\phi$ and $\psi$ for $G_1 \cong G_2$? If for some automorphisms $h \in \operato...

Consider the following family of polynomials $$P_n(X)=\sum_{i=0}^n(n+1-i)X^i,\,n\ge 1$$ Let’s write down the first few $$ \begin{align} &P_1(X)=X+2\\ &P_2(X)=X^2+2X+3\\ &P_3(X)=X^3+2X^2+3X+4\\ &P_4(X)=X^4+2X^3+3X^2+4X+5 \end{align} $$ My claim is that this family is a family of irreducible polyno...

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...