共 345 个问题，第 3/18 页
$2$ Questions in the proof of Theorem $7.18$ of Clader and Ross Begining in Algebraic Geometry
I am self studying algebraic Geometry from the textbook of Clader and Ross Beginning in Algebraic Geometry and I have $2$ in the proof of the theorem $7.18$ given on the page $201-203$. Background: define $I_a= \{F\in K[X]\mid F(a)=0\} \subseteq K[X]$. On page $201$ we see that $I_a$ can be...
Questions in Proposition $7.23$ of Textbook beginning in Algebraic Geometry by Clader and Ross
I was self studying Algebraic Geometry from the textbook of Algebraic Geometry by Clader and Ross: Beginning in Algebraic Geometry. I have questions on page $206-207$ of the textbook. Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization...
Smallest Prime Divisor of a Recursive Sequence
Let the sequence $(a_n)$ be defined by $$ a_1=3,\qquad a_{n+1}=a_n^2-2 $$ for every positive integer $n$. Let $p_n$ be the smallest prime divisor of $a_n$. Prove that $$ p_n\ge 2n+3 $$ for every $n\ge 2$. Using mathematical induction, we prove that $ a_n=\alpha^{2^{n-1}}+\beta^{2^{n-1}} $ for...
Prove that there are no positive integers $a$, $b$, and $c$ that satisfy the following equation: $a^2 + b^2 = 3c^2$
Prove that there are no positive integers $a$, $b$, and $c$ that satisfy the following equation: $$a^2 + b^2 = 3c^2.$$ Think about remainders modulo $3$ (Modular Arithmetic $\bmod 3$). What are the possible remainders of any perfect square when divided by $3$? Then, apply the method of Infinite Descent.
Is this identity, which links the Goldbach $\varphi$ function to the local count of partitions, known?
Is this identity (13), which links the Goldbach $\varphi$ function to the local count of partitions, known? https://zenodo.org/records/22048675
Affine Line over $\mathbb R$
I am reading "The Rising Sea" by Ravi Vakil. I have several question regarding section 3.2 . We know $\operatorname{Spec}\mathbb{C}[x]$ looks like, and we can associate each maximal ideal to a complex number. But what about genric point? What is the intuition behind the terminology: the elements...
For what integers $N$ does $\phi(N^2)=\phi(N)^2$?
Let $N$ be a positive integer and suppose that \begin{equation*} \phi(N^2)=\phi(N)^2 \end{equation*} where $\phi$ is the Euler function. What can we say about $N$? Here is my attempt: Of course, this equality holds for the trivial case $N=1$ because $\phi(1)=1$. If $N>1$ is a prime integer, the...
Asymptotic with sharp error term of $\sum_{p} \frac{\log p}{p^2} e^{-x/p^2}$ as $x\to\infty$
Define $$S(x)=\sum_{p} \frac{\log p}{p^2} e^{-x/p^2}$$ where p denotes prime. I need asymptotic expansion of $S(x)$ with sharp error term as $x\to\infty$. Define $v(t)=\sum_{p\leq t} \log p$, then by Euler summation we have $$S(x)=\lim_{T\to\infty}\left(v(T)\frac{e^{-x/T^2}}{T^2}+2\int_2^T...
Are the coefficients of li(x)’s asymptotic expansion optimal among approximants of the form x.P(1/log x)?
States the class — x·P(1/L), L = log x — and li's expansion x Σ (k−1)!/L^k. 2. Question 1: is it standard that those coefficients are the unique optimum, and is there a canonical reference? 3. Shows why you're asking: your π_g, your π_h^(N), the observation that x^{1/n} for n≥2 is O(√x) and...
Did Hermite solve the quintic equation by canceling weights to form an invariant variable via modular transformations?
I am trying to understand the deep mechanism behind Charles Hermite's solution to the general quintic equation using elliptic modular functions. As I understand it, under the 12 modular transformations of order 5 (associated with the modular equation of degree 6), the relevant modular forms...
Checking regularity of field extension for Chatzidakis notes
I have been referencing Chatzidakis' notes on psuedofinite fields section 6.7, found here, and her explanation of the simple case of Duret's result that the theory of any pseudo-algebraically closed fields which are not separably closed has the independence property. Let $F$ be a pseudofinite...
Does $ x_1^2 + x_2^2 + x_3^3 + x_4^3 = y_1^2 + y_2^2 + y_3^3 + y_4^3 $ have infinitely many solutions?
This question is inspired by another post of a similar question. In particular, the question 2 in the first section is looking for solutions of $ x_1^4 + x_2^4 + x_3^8 + x_4^8 = y_1^4 + y_2^4 + y_3^8 + y_4^8 $. I am wondering about the same problem, but with smaller powers. For example, what...
How to show that $5$ divides $n^2-1$ if $5$ divides $1+2n^2+3m^2$?
This question was part of my number theory assignment and I am not able to make any significant progress on it. Question: Let $n$ and $m$ be integers such that $5$ divides $1+2n^2+3m^2$. Show that $5$ divides $n^2-1$. Assume $2n^2+3m^2\equiv -1 \pmod 5$. I am not able to think which result I...
Does there exists a positive integer $n $ such that the decimal representation of $3^n$ ...
This question was asked in a masters entrance examination and I am not able to make any significant progress on this problem. Question: Does there exists a positive integer $n$ such that the decimal representation of $3^n$ starts with the digit 2019? Justify your assertion. I have been following...
Zero dimensional subscheme and section of a sheaf on $\mathbb P^2$
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...
complete intersection on $\mathbb{P}^1\times\mathbb{P}^1$?
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...
Interference-like density striations in a superposition of shifted prime-counting step functions — is there a physical interpretation?
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...
When does a non-empty CRT residue set meet a short interval?
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\}....
A continued fraction for Baxter's four-coloring constant
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...
Why can a prime-gap trajectory converge to a future value before that value appears as a prime?
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...