I find it easiest to explain the motivation as a "game" or task: you are given an arbitrary (but guaranteded to be valid) subsequence of the characteristic function of primes, so just '1's and '0's, one after another, and your task is to identify the numbers they represent. To simplify, let's...

I am an independent researcher investigating the arithmetic and structural properties of prime distributions formulated within the twin-track lattice of $6n \pm 1$. I would like to inquire about a potential algebraic and geometric invariant regarding Goldbach pairs. Consider the following model...

I read little bit about Special Relativity and there was one moment that I can't understand. It was about that there was two irreducible polynomials that have common roots: I was confused because each of these polynomials have different variables. My question is: is it possible that two...

Determine all monic polynomials $P(x)$ with integer coefficients for which there exists a monic polynomial $Q(x)$ with integer coefficients such that, for every pair of positive integers $m,n$, $P(m^2+mn+n^2)\ne 0$ and $$ P(m^2+mn+n^2)\mid Q(m^4+m^2n^2+n^4). $$ Let $$ A=m^2+mn+n^2,\qquad...

(I'm sorry for my English.) Hello. I have been reading B.Mazur's article "An introduction to the deformation theory of Galois representations". I'm at the proof of proposition1 in §30, where he discusses I-ordinary deformation. Let me write down the settings : $A$ : a Noetherian local ring which...

I am self learning Algebraic Geometry from Daniel Perrin's Algebraic Geometry textbook. I have a question on last paragraph of page $41$. Let $D(f)$ be the set of points where the function doesn't vanish and $\Gamma(V)= k[X_1,...,X_n]/I(V)$ where $k$ is a commutative field. Let $r$ denote the...

I am reading up on some algebraic geometry and came across the above definition of sheaves. Some things confuse me. When the author says in 2) of Definition 4.1, "For each inclusion of open sets $V \subset U$..", does he mean that $V$ is an open set in $X$ or in the induced topology on $U$? In...

Problem Statement Let $f: X \to Y$ be a surjective morphism of finite type between affine Noetherian schemes, where $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} A$. Suppose that for every closed point $y \in Y$ and for every $x \in X_y$ that is closed in $X_y$, the stalk map...

Let $k$ be an infinite field. Definition 1. ( Affine Hilbert function ). Let $R := k[x_1, \dots ,x_n]$ be a polynomial ring which can be viewed as a vector space over $k$. Let $R_{\le s} := k[x_1, \dots, x_n]_{\le s} $ denote the set of polynomials of total degree $\le s$ in $R$. Note that...

This question is from Proposition $6.3$ of the textbook Algebraic geometry by Daniel Perrin( Page 32). Here $k$ is a commutative field.Let $V$ be a projective algebraic set and consider a homogeneous element $f \in \Gamma_h(V)= k[X_0,...,X_n]/I_p(V)$ of degree>0. $I_p(V)$ is the ideal of...

Let $s \ge 4$ be a composite integer with $s \ne 0 \pmod3$. Computationally, for every such $s \le 2000$, I can find a prime $p$ and integer $k \ge 1$ such that $p^k - s$ is prime (usually with $p \in \{2,3\}$ and small $k$). Heuristically this seems unsurprising: for fixed small $p$, the values...

I'm trying to see that the Hecke algebra defined as $\mathbb{Q}[\text{GL}_2(\mathbb{Z}_p)\backslash \text{GL}_2(\mathbb{Q}_p)/\text{GL}_2(\mathbb{Z}_p)]$ maps to the ring of correspondences $\text{Corr}_\sim^0(M_n,M_n)$ where $M_n$ is the modular curve of elliptic curves with full $n$-torsion...

I am self studying Algebraic Geometry from the Clader and Ross Algebraic geometry textbook: Beginnings in Algebraic Geometry. This proof is given on page 275 of the textbook and I am quite confused about it. Please help me. Theorem 9.53 (Projective Nullstellensatz) Assume that $K$ is...

For instance we have the projective space itself is a manifold, and we often talk about zero set of polynomials over the projective space in algebraic geometry. So, how do the non trivial manifold properties /algebraic topology play into the study of algebraic geometry?

It is well known that for a finite field $K$ of characteristic $p$ the ring of $n$-truncated Witt vectors $W_n(K)$ is used to classify field extensions of $K$ of degree $p^n$; for details see e.g. Bosch's Algebra, chapter 4.10 on general Kummer theory. Basically the upshot is, cyclic subgroups...

I. Manifolds A non-singular homogeneous polynomial of degree $n+2$ with $n+2$ variables is a compact Calabi-Yau manifold, some of which important in string theory. For $n=2,3,5$ dimensions, we have, $$x_1^4+x_2^4+x_3^4+x_4^4 = 0$$ $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5 = 0$$...

This is different from asking whether the Prime Number Theorem implies the asymptotic. I am specifically asking whether this asymptotic has an explicit published reference (journal article, book, or monograph), rather than whether it follows from standard methods. I am looking for a literature...

Let $b\ge 2$. Partition $[0,1)$ into the $b$ equal half-open intervals $$ I_j=\left[\frac{j}{b},\frac{j+1}{b}\right), \qquad 0\le j\le b-1. $$ Define the same-bin indicator $$ H_b(x,y)= \begin{cases} 1, & x,y\text{ lie in the same }I_j,\\ 0, & \text{otherwise}, \end{cases} $$ and the centered...

Let $a(b,n)$ be the number of integer tuples $(x_1, x_2, ..., x_{k+1})$ where $0 \leq x_i \leq b-1$, such that $|x_i - x_{i+1}| = d_i$ for all $i$, where $(d_1, d_2, ..., d_k)$ are digits of $n$ in base $b$. Related patterns in this specific sequence are discussed here and here. Now consider the...

For the Fermat curve $x^4+y^4=z^4$, what does its L-function look like? I know its zeta function over prime $p$ should have the form $\frac{P_p(t)}{(1-t)(1-pt)}$, with $P_p$ a polynomial of degree $6$. The L-function should be $\prod_p P_p(t)^{-1}$, right? I think the only possible bad primes...