How can I show that every Projective variety in $\mathbb{P}^n$ is compact in the induced Euclidean topology? Should I consider, as customary, an arbitrary open cover in $\mathbb{P}^n$, and perhaps use the projection map $\pi:\mathbb{C}^{n+1}\setminus\{0\}\rightarrow \mathbb{P}^n$ which defines...

Let $p_k$ denote the $k$-th prime number, and let $\operatorname{rev}(x)$ denote the decimal digit-reversal of a positive integer $x$. Define the consecutive-prime geometric sums: $$A_n = \sum_{i=0}^{p_n} p_{n+1}^i = \frac{p_{n+1}^{p_n + 1} - 1}{p_{n+1} - 1}, \qquad B_n = \sum_{i=0}^{p_n}...

I have been investigating an empirical variant of the classical Josephus problem and would like to know whether it has been studied previously. Let $J(n,k)$ denote the survivor of the classical Josephus problem with population size $n$ and elimination interval $k$. For fixed $n$, instead of...

Let $G$ be a reductive group over a field $k$. What actually does it mean to say that $G$ is of type $A_n, B_n,\dots,G_2,{}^2A_n, {}^3D_4,...$? In case it helps, I know what the Dynkin diagrams of types $A_n, B_n,\dots,G_2$ are (but not those of types ${}^2A_n, {}^3D_4,...$). I also know how to...

This statement is given as application of earlier results on the page $22 $ of the Daniel Perrin's Algebraic Geometry textbook from which I am self studying. Here $k$ is a commutative field and $V$ is a affine algebraic set. Application $6.9$ The morphism $\phi: k \to V=V(Y^2-X^3)$ given by...

Consider the sequence defined by $a_1 = 1$ and the recurrence relation for $n \ge 2$: $$a_n = \sum_{k=1}^{n-1} a_{\gcd(n,k)}$$ Grouping the terms by their divisors $d = \gcd(n,k)$, the number of integers $k < n$ such that $\gcd(n,k) = d$ is given by $\varphi(n/d)$, where $\varphi$ is Euler's...

Background & Definition In lattice geometry, two points $A, B \in \mathbb{Z}^2$ are said to be visible to one another if the open line segment between them contains no other lattice points. Equivalently, if $A = (x_A, y_A)$ and $B = (x_B, y_B)$, they are visible if $\gcd(|x_A - x_B|, |y_A -...

I have been analyzing a recursive sequence based on the greatest common divisor that acts as a dynamic sieve for twin primes. It shares structural similarities with Rowland's prime-generating sequence but targets the difference of squares. For any integer $n \ge 2$, define the sequence...

The Fermat quintic threefold is given by the equation, $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5=0\qquad\qquad$$ $\hskip1.5in$ (Incidentally, this threefold is a Calabi-Yau manifold, a type of manifold important to string theory.) In the integers, there are only four primitive solutions known, two which...

Let $f\in S_{k+1/2}^+(4q)$ be a newform in Kohnen’s space for $q$ an odd, square-free integer. For simplicity, assume that $k$ is even. Let $$f(z) = \sum_{\substack{n\geq 1\\ n\equiv 0,1\mod 4}}a_f(n)e(nz)$$ denote the Fourier expansion of $f$ at the cusp $\infty$. Let $D>0$ be a fundamental...

I am confused about the construction of the Hecke operators. I am defining them on $\Gamma_{1}(N)$ as $$(T_{m}f)(z) = \sum_{\substack{a,d \ge 1 \\ ad = m}}\langle a\rangle\left[\Gamma_{1}(N)\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\Gamma_{1}(N)\right]_{k}f(z),$$ where $\langle a \rangle$ is...

For $U \to X$ étale, define a presheaf by $\mathcal{O}_{X_{\text{ét}}}(U) = \Gamma(U,\mathcal{O}_{U})$. I want to show that this is a sheaf on $X_{\text{ét}}$. Clearly, the sheaf condition holds for all Zariski open coverings, so it is sufficient to show the sheaf condition holds for étale...

Let $k$ be an algebraically closed field and $V$ be an affine variety. From Page 19 of Daniel Perrin’s Algebraic geometry. Lemma: Let $F,G\in k [X,Y]$ be non-zero polynomials without common factors, there is a non-zero polynomial $d\in k[X]$ and polynomials $A,B\in k[X,Y]$ such that $d= AF+BG$...

I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$...

I am self studying Algebraic Geometry from the Daniel Perrin's textbook. $k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$ is the...

Let $$B:=\{n\geq 13: n \text{ odd and there exists a prime } q \text{ such that } 9n^2+1-q \text{ is prime, where either } q=3, \text{or } q\geq 11,q \equiv 2\pmod 3, q-1 \text{ is not a square}\}$$ I am trying to understand whether the following “almost-all Goldbach” statement is known /...

I am exploring a family of integer sequences $(k_n)$ that combine explosive quadratic growth with specific "modular" reduction rules based on the proximity to perfect squares and powers of two. I’ve categorized these reduction mechanisms as "Valves." The Core Growth Function: Let $f(x) = ax^2 +...

Let $(a,b,c)$ be positive integers such that a $abc+1 \mid a^2+b^2+c^2$. Then $\dfrac{a^2+b^2+c^2}{abc+1}$ can be written as the sum of 2 positive squares. Proposed by Sam Vandervelde.

Professor Manin's 1990 ICM talk states there is a convincing case to be made that these groups are, in some sense, "more fundamental" in number theory than even the integers. The talk's purpose being a broadest-possible survey of the works of Professor Drinfel'd, Professor Manin does not deem it...