MathOverflow

Are there some $P,Q \in \mathbb R_+[x]$ with $(x+10)^{2025}=(x+2025)^2P(x)+(x+2024)^2Q(x)$ ?
PS : the AI give an negative answer in the case $(x+1)^{2025}$
I have posted the question here (*), but no answer.

Let $k$ be fixed. Let $q_1,q_2,\dotsc,q_k$ be coprime with product $\prod_{j=1}^k q_j$ of size about $N$. Let $n$ range over integers in $[1,\sqrt{N}]$ coprime to $q_1,q_2,\dotsc,q_k$. Is the vector $$\left(\frac{n}{q_1}\right),\left(\frac{n}{q_2}\right...

In my research, I have come across a particularly nasty integral. Let $a$ and $\delta$ be such that $-1 \le a+\delta\cos(\psi) \le 1$ for all $\psi \in [0,\pi]$ and $\delta>0$. I would like to have an upper bound on the integral
$$\int_{0}^{\pi}\frac{d\psi}{1+\left|\frac{\arccos(a+\delta \cos(\psi))}{\epsilon}\right|^k}$$for integer $k>2$ and $\epsilon\le 1$. Since this is a step in a longer proof I would really like the bound to depend on $a,\delta, \epsilon$, and $k$ if...

Let $G$ be a discrete group. Since I'll be using the term in a more general way than is usual, let me spell out what I mean by a $K(G,1)$: it is a pointed space $(X,x_0)$ with the following three properties:
$\pi_1(X,x_0) \cong G$.
The space $X$ is Hausdorff, path connected, locally path connected, and semilocally simply connected. It therefore makes sense to talk about the universal cover $\widetilde{X}$ of $X$....