Questions tagged [nt.number-theory]
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
15,905
questions
6
votes
0
answers
662
views
Show this number always is composite number
Conjecture:
Let $m$ be a positive integer. Then
$$f(m)=(2m)^{2m+1}+m^{2m+1}\cdot (2m+1)^m+(2m+1)^{2m}$$
is not a prime number.
One can prove it when $m$ is odd number, it is clear that $f(m)$ ...
5
votes
0
answers
353
views
Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?
Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so:
\begin{aligned}
\zeta_H(s)
:=
tr( H^{-s} )
\\
:=
\sum^\infty_{n=1} ...
4
votes
0
answers
205
views
Explicit elements of the first cohomology of modular curves
Let $M$ be a modular curve and $\pi:E\to M$ the universal elliptic curve. For a prime $\ell$, let $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. I am wondering whether there are any explicit ...
4
votes
1
answer
445
views
Moments of merit
The merit of a prime gap equals $(p_{n+1}-p_n)/\ln p_n$. One can interrogate the statistics of merit by first restricting $n<M$ for some $M$, and then letting $M$ approach $\infty$. The very ...
7
votes
1
answer
355
views
index of smooth varieties
What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$?
EDIT: Also assume that the residue ...
4
votes
2
answers
887
views
Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya
I am reading the paper "$p$-adic cohomology: from theory to practice" by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack ...
3
votes
1
answer
384
views
Moduli problem of stable nodal curves over the integers
Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...
2
votes
0
answers
166
views
Möbius function in every short interval
I am looking for references on conjectures or heuristics concerning cancellations of the Möbius function in very short intervals, namely how small is it believed one can take $H$ so that
$$\sum_{X \...
9
votes
1
answer
458
views
Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
8
votes
2
answers
938
views
How to compute Dedekind eta function efficiently?
According to wiki: https://en.wikipedia.org/wiki/Dedekind_eta_function, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic ...
3
votes
1
answer
418
views
The degree of the cube root of the $j$-invariant
I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...
17
votes
1
answer
748
views
Congruences Ramanujan-style
Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by
$$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$
The numbers $p_t(n)$ can be regarded as enumerating ...
3
votes
1
answer
167
views
Diophantine equations and 'quasi-paucity'
Let $X,Y \geq 1$. I am interested in the number of solutions of the following diophantine equations:
$$S_1\colon \, \, x_1y_1^3 = x_2 y_2^3 $$
Let $N_1(X,Y) $ denote the number of solutions to $S_1$ ...
1
vote
1
answer
137
views
How to compute the Müller modular polynomials?
According to R.A.Kazmi's dissertation "Isogenies and Cryptography" (Page 22), given an isogeny degree $l$, the Müller modular polynomials are defined as
$$G_l(x,y)=\sum_{r=0}^{l+1}\sum_{k=0}^{v}a_{...
15
votes
3
answers
1k
views
Perfect numbers and perfect powers
This was asked earlier at MSE.
The observation that 28 = 27 + 1 shows that it is possible to have consecutive perfect numbers and perfect powers. However, this must be extremely rare. Is it ...
0
votes
1
answer
140
views
Searching for matrices with some property
I don't know if this question is considered research-related. If not, I will move it to Math SE.
I am searching for matrices with the property
$$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) ...
1
vote
1
answer
308
views
A product of polynomials
Let $f(n)=1+x^n+x^{2n}+...+x^{n^2}.$
Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers.
Let $a(n)$ be the sequence of integers such that the coefficients of the ...
4
votes
0
answers
190
views
Can one define certain 'interesting' topology for $\overline{\mathbb{Q}}$ on which the absolute Galois group $G_\mathbb{Q}$ acts continuously?
The story can be generalized to the case of projective variety, however, we consider the most basic case.
Fix an embeding (i.e.'complex place') $\iota:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$, ...
3
votes
0
answers
187
views
Growth Rate of the Square-Free Part
In the course of considering this Diophantine equation, I convinced myself that the following question is interesting:
If $n$ is large, must it be the case that the square-free part of $2^n-1$ is ...
2
votes
0
answers
114
views
Sieving the values of an arithmetic sequence which is infinitely many times $1$
I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
5
votes
0
answers
113
views
$m$-thick sets with small $n$-fold sumsets in finite cyclic groups
Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
9
votes
2
answers
362
views
Consecutive polynomial non-residues modulo a prime
Given a polynomial $f(x)\in \mathbb{Z}[x]$ of degree at least 2 and a positive integer $t$, does there always exist infinitely many primes $p$ such that the range of $f(x)$ modulo $p$ does not contain ...
4
votes
0
answers
189
views
Restricted Iwasawa theory
Let $p$ be a prime number, let $K$ be a number field, and let $L$ be a Galois extension of $K$ such that the Galois group $\Gamma$ of $L$ over $K$ is (continuously) isomorphic to the (additive) group $...
5
votes
1
answer
212
views
Positive polynomial equations mod integers
Let $f(x)$ and $g(x)$ be polynomials with integer coefficients such that $f(x)>0$ and $g(x)>0$ for all real values of $x$. Suppose that for every integer $n$, if $f(x)=0$ has a solution mod $n$, ...
18
votes
1
answer
674
views
Arithmetic motivations for modularity in higher rank
The classical setting of modularity is that one can associate elliptic modular forms (or automorphic representations of GL(2)/$\mathbb Q$) to elliptic curves over $\mathbb Q$. This has far-reaching ...
2
votes
2
answers
1k
views
What is the natural density of hyper prime numbers?
What do we mean by hyper prime numbers? Well, roughly speaking they are natural numbers which are prime with respect to hyperoperators in arithmetic such as exponentiation, tetration, pentation, et ...
6
votes
0
answers
681
views
What are the fastest ways to calculate class number of number fields?
Given a number field $K$, which approaches help us to calculate the class number $h(K)$ of $K$?
I am aware that the question is broad but any argument would be helpful.
Some basic approaches I know:...
3
votes
1
answer
230
views
Questions about a product of trinomials
Let $f(n)=1+x^n+x^{2n}$
Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers.
Let $a(n)$ be the sequence of integers such that the coefficients of the series $f(a(1)) f(...
3
votes
1
answer
292
views
Degeneracy maps and cusps
Let $N$ be a positive integer and $p$ a prime not dividing $N$. Let $X_0(N)$ be the modular curve (over $\mathbb{Q}$) associated to the congruence subgroup $\Gamma_0(N)$, which is the subgroup of $\...
5
votes
2
answers
429
views
Is there any work on the Gauss circle problem over function fields? [closed]
I would be thankful if someone had references to provide...
19
votes
1
answer
883
views
What computer program for automorphic forms
This question has its origins in this entertaining discussion on MO.
There are many programs (CAS) and libraries that are able to handle algebraic expressions. These are both a verification tool for (...
3
votes
0
answers
164
views
$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property
Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...
-1
votes
1
answer
115
views
probability of m to be a primality radius of n
Disclaimer : this is a crosspost from MSE, as the question got one upvote but no comment or answer whatsoever.
Assuming Goldbach's conjecture, let's denote by $r_{ 0}(n):=\inf\{r\geq 0,(n−r,n+r)\in\...
8
votes
1
answer
502
views
Computational complexity of finding the class number
Let $f(x)$ be an integral irreducible monic polynomial and $\alpha$ be its root. What is the computational complexity of finding representatives of ideal classes of the integral ring $\mathbb{Q}[\...
4
votes
1
answer
231
views
Meromorphic continuation of local zeta integrals
Let $f$ be a Maass cusp form for $\text{SL}_2(\mathbb{Z})$ on the upper half plane. Let $\varphi_0$ be its lift to an automorphic form on $G = \text{PGL}_2(\mathbb{R})$ and let $\pi = \pi_{f} =\langle ...
8
votes
2
answers
336
views
Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?
Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
5
votes
0
answers
124
views
How good are these probabilistic algorithms for the NP-hard problem gcd of sparse polynomials?
The paper NEW NP-HARD AND NP-COMPLETE POLYNOMIAL AND INTEGER DIVISIBILITY PROBLEMS David A. PLAISTED”
defines sparse polynomial as set $\{(a_i,i)\}$ and $f=\sum a_i x^i$.
On p.5:
Theorem 3.3. The ...
2
votes
1
answer
143
views
Difference of powers
Prove or disprove: Any positive integer N can be expressed as N=a^x-b^y with a,b,x,y >1. (I don't know the answer or whether this conjecture has been addressed.)
7
votes
1
answer
269
views
Why does the class equation have rational coefficients?
Let $j$ be the Klein modular invariant and let $\Gamma=SL_2(\mathbb Z)$ be the modular group. Consider the set of primitive quadratic forms $ax^2+bxy+cy^2$ of discriminant $d<0$.
The root of a ...
1
vote
0
answers
108
views
Complex multiplication of Jacobians of quadratic twists of a fixed hyperelliptic curve
Let $C: z^2 = f(x,y)$ be a hyperelliptic curve defined over $\mathbb{Q}$, with $f$ a binary form with integer coefficients and non-zero discriminant. Let $A_C$ denote the Jacobian variety of $C$, and ...
5
votes
0
answers
432
views
Why does Faltings' Siegel lemma imply Siegel lemma?
Recall the Siegel lemma:
Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
7
votes
1
answer
638
views
How is the Eichler-Shimura congruence related to L-functions?
My understanding is that the Eichler-Shimura relation expresses the Hecke operator $T_p$ in terms of the geometric Frobenius map. Specifically, $T_p = Frob + Ver$ for Frobenius map $Frob$ and it's ...
5
votes
1
answer
519
views
Few questions regarding Heath-Brown's identity
Heath-Brown's identity states: Let $K \geq 1, z \geq 1.$ Then for any $n < 2 z^K$ we have
$$
\Lambda(n) = - \sum_{1 \leq k \leq K} (-1)^k {{K}\choose{k}}
\sum_{ \substack{ m_1 \cdots m_k n_1 \cdots ...
12
votes
0
answers
309
views
Writing integers as a product of as few elements of $\{\frac21, \frac32, \frac43, \frac54, \ldots\}$ as possible
Is the number of elements we need to construct $x$ equal to $\log_2(x) + O(1)$?
This question is inspired by question 2 of the 2018 European Girls' Mathematical Olympiad. I previously posted it on ...
8
votes
1
answer
764
views
Tate modules of commutative group schemes over finite field
Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\...
12
votes
2
answers
989
views
Polynomials dense with primes
Let $p(n)$ be a polynomial with integer coefficients.
Define $\Delta( p(n) )$, the prime density of $p(n)$, to be
the limit of the ratio with respect to $n$
of the number of primes $p(k)$ generated ...
5
votes
1
answer
457
views
Growth order of numbers whose prime factors are all congruent to +1 or -1 modulo 8
Here are the numbers whose prime factors all congruent to $\pm 1\pmod 8$:
http://oeis.org/A058529
My questions are:
(1) What is the order of growth of these numbers? That is, what is the order of ...
8
votes
0
answers
295
views
Finding a cyclic cubic extension of a field
Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...
15
votes
1
answer
530
views
Counting primitive lattice points
In Lemma 2 of [1], Heath-Brown proves the following (I state a simplified version of a more general result):
Let $\Lambda \subset \mathbb{Z}^2$ be a lattice of determinant $d(\Lambda)$. Then
$$\# ...
10
votes
1
answer
483
views
Estimating volumes in the trace formula
Let $G$ be a reductive group. In many instances of the trace formula, elliptic terms corresponding to the $G(k)$-conjugacy class of $\gamma \in G(k)$ are weighted by the following volumes
$$v_\gamma = ...