I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław Sierpiński's 250 Problems in Elementary Number Theory, I came across a remark stating that A. Mąkowski...

I ran a simulation where for each 2-digit combination with 30 symbols (so 0 to T), it checked, from base 2 to base 10,000, in how many bases that specific symbol combination resulted in a prime number. The top 10 were 65,6B,6H,6T,6N,61,6D,67,6J, and 6P. All starting with 6. Anyone have any idea...

This problem is from the book Álgebra comutativa em quatro movimentos by Borges and Tengan. Let $k$ be a field of characteristic not equal to $2$ or $3$. Let $f(x, y) = y^2 - x^3 - 10$ and consider the ring $R = k[x, y]/(f(x, y))$. Show that $R$ is a Dedekind domain. Edit. We also have to assume...

I am reading the paper Oscillatory Integrals with Uniformity in Parameters by Kiral, Petrow, and Young, and I am having trouble understanding the notion of an inert function introduced in Definition $2.1$. These are my confusions Since $X=X_T \in [1,\infty]$.Then if we consider a family of...

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly quotients), but I will restrict to two. Compute the Krull dimension of the following rings. $R =...

$\newcommand{\e}[1]{\exp\left(#1\right)}$ $\newcommand{\le}{\leqslant}$ In Terry Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function), Exercise 5 outlines the derivation of the Vinogradov-Korobov bound for Dirichlet $L$-functions. Let $\chi$ be a non-principal character of...

Let P(n) and Q(n) be two non-trivial polynomials in n with rational coefficients and z[P, Q] is the value of infinite sum of P(n)/Q(n) from n=1 to +∞ (only when it converges, in which the degree of Q should be larger than or equal to the degree of P plus 2). Claim: It is impossible for z[P,Q] to...

Let $C$ be a plane curve and suppose, for simplicity, it has a singularity at the origin. Let $\tilde{C}$ be it's strict transform after blowing up $(0,0).$ Is it possible for $\tilde{C}$ to have a singular point not at the origin?

These days I've been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on infinite root stacks. At the moment I'm looking at their proposition 4.9....

I mean, if a function $F(s)$ is in Selberg class, and $F(s)$ has a pole of order m at $s=1$, is it true that there exists a function $G(s)$ in Selberg class such that $F(s)=\zeta^m(s)G(s)$, and $G$ is entire?

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use the irreducible components to say something about the annihilator of $f$ in $R$? (Perhaps it...

I found this problem where I think there is a mistake but am not 100% sure: Let $Z$ be a closed subset of a topological space $X$. If $Z$ is irreducible we call codim($Z,X$) as the supremum of lengths of the chains of irreducible closed subsets of $X$ which contains $Z$: $$Z\subset Z_0\subsetneq...

One of the axioms for Selberg class is that for some natural number $m$, $(s-1)^mF(s)$ extends to an entire function of finite order. What is the use of "finite order"? If we allow it to be of infinite order, what will happen?

Elliptic curves have L-functions that correspond to modular forms. Elliptic curves are degree 3 algebraic curves. I want to know if quadratic curves have L-functions. If they do, are these L-functions related to modular forms?

I am currently studying algebraic geometry and trying to understand projective spaces. let $S = \mathbb{Z}[x_0, \dots, x_n]$ so that $\mathbb{P}^n_{\mathbb{Z}} = \operatorname{Proj}(S)$. I read that if we remove the standard open affine subset $D_+(x_i) = \{ \mathfrak{p} \in...

I'm studying algebraic geometry and have a question on Kahler differentials. Let $k$ be a commutative ring with unit and $A$ a $k$-algebra. I already have the intuition if we picture $A$ as some ring of functions on $\operatorname{Spec}{A}$, the the module of relative Kahler differentials...

I'm self studying commutative algebra. Here is the question: Edit(Background and Definition): Let $k$ be a field, we define an affine variety $V\subseteq k^n$ as the set of common zeroes of some polynomials $f_1,\dots, f_m\in k[x_1,\dots, x_n]$, define its coordinate ring as $k[x_1,\dots,...

Given a real polynomial $f(x)=a_0x^d+\cdots+a_1 x$, I want to give a sharp estimate for $\sum_{n\le X} e(f(n))$. If $f(x)=ax$ is a linear function, we have $$\sum_{n\le X} e(\alpha n)\ll \min (X,\|\alpha\|^{-1});$$ If $f(x)\in\mathbb{Z}[x]$ where $p$ is a prime, using Weil's bound for...

https://en.wikipedia.org/wiki/Rational_mapping The usual example is that $ \mathbb {P} _{k}^{2} $ is birational to the variety $ X $ contained in $ \mathbb {P} _{k}^{3} $ consisting of the set of projective points $ [w:x:y:z] $ such that $ xy-wz=0 $, but not isomorphic. Indeed, any two lines in...

This question follows from Lemma 6.1.1 from CH 6.1 of Field Arithmetic by Fried and Jarden. It is the subsection on Decomposition groups.' The following paragraph sets up the notation used. In the construction of Decomposition groups the chapter starts by defining $R$ to be an integrally closed...