Let $\mathbb Z^+$ be the set of positive integers. In 1934, Romanoff proved that$$\liminf_{x\to+\infty}\frac{|\{n\le x:\ 2n+1=p+2^k\ \text{for some prime}\ p\ \text{and}\ k\in\mathbb Z^+\}|}x>0.$$...

In a paper published in 1971, R. Crocker proved that there are infinitely many positive odd numbers not of the form $p+2^a+2^b$ with $p$ prime and $a,b\in\mathbb Z^+=\{1,2,3,\ldots\}$. The proof makes ...

Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...

MotivationAlternative proof of fullness of collection on $D^b(\mathbb P^n)$Richard Thomas's notes on HPD (page 2) mention an alternative proof (rather than Beilinson's resolution of the diagonal) ...

The precise conjecture that I want to prove or disprove is the following:Let $P_n$ denote the path graph with n vertices, $S_n$ the star graph with n vertices, and $T_n$ an arbitrary connected tree ...

In a paper published in 1985, Shih-Ping Tung observed that an integer $m$ is nonzero if and only if $m=(2x+1)(3y+1)$ for some $x,y\in\mathbb Z$. In fact, we can write a nonzero integer $m$ as$\pm3^a(...

Theorem 5.1.2 isLet $U : Grp\to Set$ be the forgetful functor. If $H : J \to Grp$is such that the composite $UH$ has a limit $L$ and a limiting cone $\nu : L \to UH$in $Set$, then there is exactly ...

I am studying fixed-point equations of the form $y^k = y$ ($k \in \mathbb{N}$) in $\mathbb{Z}_n$, where here $\mathbb{Z}_n$ denotes the ring of $n$-adic integers (i.e., the projective limit $\...

Let $X \subset \mathbb{P}^n$ be a complex projective variety with rational singularities, and $Y$ a smooth subvariety in $\mathbb{P}^n$ that meets $X$ transversally, that is to say their tangent ...

Let $\mathcal{B}(n,w)$ be the set of all binary vectors of length $n$ and constant weight $w$, i.e.,$\mathcal{B}(n,w) = \{ x \in \{0,1\}^n : \mathrm{wt}(x) = w \}$.The Hamming ball of radius $r$ (in ...

Let $i_{\ast}, i^{\ast}: Sh(X) \to Sh(Y)$ be a geometric morphism of topos.In the derived category $D(X)$ of abelian sheaves on $X$, we can consider the internal derived Hom: $R\mathcal{Hom}_{D(X)}(F,...

$2^n$ players $P_1, \dots, P_{2^n}$, ordered in decreasing order of skill are placed uniformly at random at the leaves of a binary tree of depth $n$.They play a knockout tournament according to the ...

Consider the following PDE:$$-\Delta u + \alpha u + \beta (x \cdot \nabla) u = 0.$$Is there any nonzero weak solution of this equation on $\Bbb R^n$, in $H^1$ or other function spaces, for some ...

Let $G$ be a finite group and $Sub(G)$ denote the number of subgroups of $G$ including the trivial subgroup and $G$ itself. I believe the following is true:$Sub(H \times K)\leq Sub(H \rtimes K)$ for ...

I encountered this mathematical structure but need helpdecoding the notation:[T = \prod_{i=1}^{\infty} \mathbb{N}]with hierarchy levels:$!1, !2, !3\ldots$$T1G, T1H, T1B\ldots$$A1, A2, A3\...