Let $E$ be a vector bundle of rank $2$ on $\mathbb P^2$ and $Z$ be zero dimensional subscheme of length $k$ supported on a point $p \in \mathbb P^2$. What is the number $h^0(E(m) \times \mathscr O_Z)$ for an integer $m$? My intuition is that it should be $2k$. But it seems in some literature it...

Let $S=k[x_0,x_1;y_0,y_1]$ be the bihomogeneous coordinate ring of $\mathbb{P}^1\times\mathbb{P}^1$. Suppose that $F\in S_{(d_1,d_2)}$ is a generic bihomogeneous form of bidegree $(d_1,d_2)$, and let $F_1\in S_{(a_1,b_1)}$, $F_2\in S_{(a_2,b_2)}$ be generic forms of lower bidegrees. The question...

I'm looking at the family of step functions $\Phi_p(x) = \pi(2x - p)$, where $\pi$ is the prime-counting function and $p$ ranges over the primes. When I superimpose these functions for many values of $p$ (see attached figure, $n$ up to $5 \times 10^7$), the result is a dense region with internal...

Let $m_1,\ldots,m_s$ be pairwise coprime positive integers, and let $$ M=\prod_{i=1}^s m_i. $$ For each $i$, let $A_i$ be a non-empty proper subset of residue classes modulo $m_i$. Define the CRT-allowed residue set $S \pmod M$ by $$ S=\{x \pmod M : x \pmod {m_i} \in A_i \text{ for every } i\}....

I found the following infinite continued fraction: $$ \operatorname{C_{B4CC}}=\cfrac{2}{1+\cfrac{2}{1+\cfrac{6}{1+\cfrac{3}{1+\cfrac{10}{1+\cfrac{4}{\ddots}}}}}} $$ where $\operatorname{C_{B4CC}}$ denotes Baxter's four-coloring constant and the partial numerators are defined by interweaving...

I am studying a deterministic construction based on the consecutive gaps between primes. The phenomenon I am interested in is NOT that arithmetic on primes sometimes produces another prime. The interesting point is that the same future value can be generated by different rows of a cumulative...

Let $x,n$ be positive integers. Consider the equation $$e^x+x^e=e^n$$ I would like to know whether this equation can have any positive integer solutions. Since $x^e \gt 0$, any solution must have $n \gt x$. Dividing by $e^x$, $$1+\left(\frac{x}{e}\right)^e=e^{n-x},$$ so $$n-x=...

I'm reading CLS's Toric Varieties right now, and something is confusing me. By Theorem 1.1.17, an affine variety $V$ is toric iff $I(V)$ is toric, i.e. prime and generated by binomials. Now the variety $V = V(X+Y-Z) \subset \mathbb{C}^3$ seems to me to be toric simply because it's isomorphic to...

I was playing around with Goldbach representations and came up with the following class of primes. Consider an even integer $n$ that can be written as a sum of two odd primes, $$ n=p+q,\qquad p\le q. $$ For a fixed $n$, define $GGT(n)$ to be the largest possible value of $q$ among all such pairs...

P.412 of volume 3 of Gauss's collected works contains two unpublished remarks of Gauss that apparently have not been discussed yet. The first one is of number-theoretic significance, while the second relates the value of $Q([a+bi]\varphi)$ at a point $\varphi=\text{arcsinlemn} (x)$ such that...

Let $p_n$ be the $n$-th prime and let $$ g_n = p_n-p_{n-1}. $$ Define $$ S_n=\sum_{i=2}^{n} g_i=p_n-2 $$ and $$ V_n=S_n+g_n=p_n+g_n-2. $$ I call a convergence the occurrence of the same value $V$ at two or more distinct positions. For example, for $V=103$: $$ 95+8=103,\qquad 99+4=103,\qquad...

I am seeking guidance on how to properly classify the singular projective surface $S \subset \mathbb{P}^3$ defined by the degree 4 homogeneous polynomial $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$, which is known to contain some elliptic curve of rank 1, alongside a unique isolated singularity at $P_0 =...

I'm studying theorem III.7.5 from Hartshorne's book. There are a few things I don't understand in this proof. Theorem. Let $X$ be a projective scheme over a field $k$. Then $X$ has a dualizing sheaf $\omega_X^\circ$. Proof. Let $\dim(X) = n$. Embed $X$ as a closed subscheme of $P:=\mathbb{P}^N$...

Let $p$ be an odd prime and let $n \ge 3$. I have been considering the following family of polynomials. Set $$ r_n:=\frac{1}{2\sqrt{n-1}}, $$ and define $$ B(p,n) := r_n+\frac{p+1}{r_n}+\frac{2p}{r_n^{n-1}}. $$ Let $$ A(p,n) := 2+2p\left( \left\lfloor \frac{B(p,n)-2}{2p} \right\rfloor+1 \right)....

Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me whose reference I need. Does there exist any better bound unconditionally? Any help will be highly...

We consider the odd-step block transitions of the Collatz mapping by collapsing intermediate even divisions: $$3n_i + 1 = 2^{k_i} n_{i+1}, \quad \text{where } k_i = v_2(3n_i + 1)$$ We model potential upward expansion chains ($1^r$, where $k_i = 1$ for $r$ consecutive steps) followed by...

Does the proposition "if f(n)=an+b has two distinct primes f(c),f(d), with a>0 then it has infinitely many primes" imply Dirichlet's Theorem?

Let $n$ be a positive integer and let $p>3$ be a prime number such that $p\mid n^2+n+1$. Prove that $(n+1)^p\equiv n^p+1\pmod{p^3}$. Set $P(x)=\dfrac{(x+1)^p-x^p-1}{p}$. First use $p\mid n^2+n+1$ to show that $n^3\equiv1\pmod p$ and hence $p\equiv1\pmod3$. Now let $\omega$ be a primitive third...

I know of Conway's chained arrow notation. I have read the Wikipedia article on it, but it still didn't give me a rigorous proof that it exists and is unique. So, to make my question precise, suppose we are given a finite nonempty sequence $S$ of positive integers. How does one rigorously define...

I have been doing a computational/visual exploration of the Riemann zeta function and its completed xi function. I am a software developer and mathematics enthusiast rather than a professional mathematician, and my main aim here is to identify the established theory behind the following...