共 299 个问题，第 1/15 页
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...
Does the equation $x^e+e^x=e^n$ have any real solutions?
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 $$n-x=...
Is $V(X+Y-Z)$ a toric variety?
I'm reading CLS's Toric Varieties right now, and something is confusing me. By Theorem 1.1.17, an affine variety $V$ is toric iff $I(V)$ is toric, i.e. prime and generated by binomials. Now the variety $V = V(X+Y-Z) \subset \mathbb{C}^3$ seems to me to be toric simply because it's isomorphic to...
Are these ‘GGT-less primes’ already known?
I was playing around with Goldbach representations and came up with the following class of primes. Consider an even integer $n$ that can be written as a sum of two odd primes, $$ n=p+q,\qquad p\le q. $$ For a fixed $n$, define $GGT(n)$ to be the largest possible value of $q$ among all such pairs...
Proving two unpublished assertions by Gauss on special values of lemniscatic functions.
P.412 of volume 3 of Gauss's collected works contains two unpublished remarks of Gauss that apparently have not been discussed yet. The first one is of number-theoretic significance, while the second relates the value of $Q([a+bi]\varphi)$ at a point $\varphi=\text{arcsinlemn} (x)$ such that...
Do repeated convergences in the prime-gap sequence contain predictive information about future primes?
Let $p_n$ be the $n$-th prime and let $$ 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...
Classify the singular projective surface $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$
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 =...
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