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
15,954
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Can each ideal class contain an ideal with norm equal to $1$?
Let $K$ be an imaginary quadratic number field. Let $\mathcal O$ be an order in $K$. Can it happen, that there are $h(\mathcal O)>1$ fractional proper $\mathcal O$-ideals, representing the ideal ...
- 2,481
22
votes
3
answers
1k
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Numbers with known finite irrationality measure greater than 2
For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that
$$ \left| \alpha-\frac{p}{q}\right|...
- 151
6
votes
0
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90
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Computing all eta quotients of given weight and level
I have written a rather naive program for finding all holomorphic eta quotients of
given weight and level (and varying character). When the level has few divisors it is
very fast, but incredibly slow ...
- 11.8k
8
votes
2
answers
633
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Misunderstanding in the hypotheses of Schlessinger's criterion
In studying deformation theory of Galois representations, I've come surely to an error, relating Schlessinger's criterion.
Let's fix a representation $\bar{\rho}$ of a group $G$ and let $D_{\bar{\...
- 60.4k
1
vote
0
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Transcendance in function fields
Denote by $\Omega$ the completion of an algebraic closure of $\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$ for the valuation $-\deg$. Let $(a_n)_n$ be a sequence of $\overline{\mathbb F_q(T)}\...
- 45.8k
7
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0
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894
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The Möbius function as eigenvalues
Let the $N$ by $N$ matrix $A$ be defined by the tetration:
$$\Large \text{If } \gcd(n, k)=1 \text{ then } A(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{\Re\left(\frac{1}{n^s}\right)}}}}}}_m \text{ else }...
- 1,133
5
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1
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480
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A problem about real quadratic field
I have calculated some real quadratic field 's Hilbert class field with class number $2$,and I found they were satisfied $Gal(H_{K}/Q)\cong Z/2Z\oplus Z/2Z$,here $H_{K}$ is the Hilbert class field of ...
- 36.2k
0
votes
0
answers
128
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A quadratic trinomial that generates only prime numbers of the form $4m+1$
It is known that Euler's polynomials $\,n^2+n+p\,$ ($p\,$ prime) represent a prime for $\,n=0,\,...,\,p-2\,$ if and only if the field $\,Q (\sqrt{1-4p})\,$ has class number $\,h=1$.
The best ...
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6
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1
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Is there a connection between the average 'compositeness' of a rational number and $\phi$ (golden ratio)?
Let $n\in N$, where $n = p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{m}^{k_{m}}$ for $p_{i}$ prime.
Define the 'density' of $n$ as:
$d(n) = \frac{(p_{1}+1)^{k_{1}}(p_{2}+1)^{k_{2}}...(p_{m}+1)^{k_{m}}}{n}$
...
- 12.7k
6
votes
1
answer
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Restriction of product of automorphic forms
Let $W \subset V$ be quadratic spaces over a number field $F$.
Let $G_n=SO(V)$ and $G_m=SO(W)$ and we consider $G_m$ as a subgoup of $G_n$ via a diagonal embedding.
Let $f$ be an automorphic form of ...
- 22.6k
0
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1
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does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]
Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
- 5,169
4
votes
1
answer
495
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How does this calculation of Siegel make sense?
I am reading Siegel's paper Zum Beweise des Starkschen Satzes. Let $K$ be an imaginary quadratic field with $d_K=-p$, $p=4k+3$ a prime, and such that $h_K=1$.
Let $f=4m+1$ be a prime inert in $K$, ...
- 32.2k
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votes
0
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82
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Mean and logarithmic values for arithmetic function
Define the mean value for function $f$ as $\lim \limits_{x \to \infty} \frac{1}{x} \sum \limits_{n \leq x} f(n)$ if the limit exists denoted as $M_f$
Define the logarithmic value for function $f$ as $...
- 60.4k
6
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146
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The valuation of finite extension of an non-archimedean field
Let $(k,|.|)$ be a non-archimedean complete field and $(L,|.|)$ be a finite extension of $(k,|.|)$, $[L:k]=n$, such that $L=k(\xi)$. Let $\phi$ the homomorphism of $k$-Banach algebra
$$\begin{array}{...
-3
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1
answer
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Equation $p\cdot q\cdot r=a^3-1+43\cdot (b^2-1)$
$p\cdot q\cdot r=a^3-1+43\cdot (b^2-1)$
p, q, r are primes.
a, b integers>0.
Is this equation a Mordell equation?
Has this equation infinitely many solutions?
- 53.6k
9
votes
1
answer
335
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The $S$-unit equation for functions on curves
Let $X$ be a smooth projective connected curve over a number field $k$, and let $S \neq \emptyset$ be a finite set of closed points of $X$. The curve $Y = X \setminus S$ is affine, and we denote by $R$...
- 9,417
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Has the equation $p^3-q^2+2=2^3\cdot q$ infinitely many solutions for $p$ and $q$ prime? [closed]
Consider the equation:
$p^3-q^2+2=8\cdot q$
Has this equation infinitely many solutions for $p$ and $q$ prime?
- 99.2k
1
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3
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233
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Perfect squares between certain divisors of a number
Let $n$ be a positive integer. We will call a divisor $d(<\sqrt{n})$ of $n$ special if there exists no perfect squares between $d$ and $\frac{n}{d}$. Prove that $n$ can have at-most one special ...
- 30k
7
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2
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509
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Conjecture about an exponential sum
Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum_{x \in ...
- 60.4k
1
vote
1
answer
189
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Resolution of an inequality on integers
I’m trying to resolve respect to $k$ the following inequality,
$$
k\left(\log k +\log \log k-\alpha+O\left(\frac{\log \log k}{\log k}\right)\right)\geq x,
$$
in order to obtain, under the condition $...
- 118k
3
votes
1
answer
237
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Is this Siegel's formula correct?
In the paper Zum Beweise des Starkschen Satzes Siegel considers the function
$$L_q(s)=\sum_{n=1}^{\infty}\left(\frac{q}{n}\right)n^{-s},$$
where $q$ is a discriminant of a quadratic number field ...
- 32.2k
1
vote
0
answers
143
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About the distribution of Fibonacci numbers that are primes
Let's consider the Fibonacci sequence, that is the sequence of naturals defined by:
$F_1=F_2=1$
$F_{n+1}=F_{n}+F_{n-1}$
It is an open problem whether the sequence contains an infinite number of ...
- 1,000
7
votes
1
answer
141
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How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?
I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason?
https://oeis.org/A002487 : Stern's diatomic series
https://oeis.org/...
- 49.5k
3
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0
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80
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Minimizing weight values while preserving the collection of maximum-weight independent sets
Consider an undirected graph $G = (V,E)$ with a weight function $w \colon V \to \mathbb{N}$ on its vertices. Let $\alpha_w(G)$ denote the maximum weight of an independent set in $G$, i.e., the maximum ...
- 291
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Modern reference for Andre Weil's 'Sur les courbes algébriques et les variétés qui s'en déduisent'
I'm currently interested in the cardinality of the set of values of a polynomial over a finite field.
I found a paper
Saburo Uchiyama, Sur le Nombre des Valeurs Distinctes d'un Polynome a ...
- 33.9k
2
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0
answers
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Reference/PDF request for paper by Sathe and related note
I am looking for a PDF version of the following articles:
Sathe, L. G. - On a problem of Hardy on the distribution of
integers having a given number of prime factors, (I. - IV.) J. Indian Math. Soc. ...
- 549
11
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3
answers
1k
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References for general Hasse-Weil zeta function
Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case.
I am ...
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4
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0
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143
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Moments of the prime counting function given the moments of the second Chebyshev function
I have read this article (Montgomery and Soundararajan: Primes in short intervals. http://arxiv.org/abs/math/0409258 ). In the second page of the article, it is stated that the mean and variance of $\...
- 60.4k
7
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1
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400
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A variant of Lambert function
How to express the solution of $x^{x+1}=a$ using Lambert function? I know that the standard Lambert function can be used to describe the solution of $x^x=a$. I wonder if $x^{x+1}=a$ can be addressed ...
- 11.9k
-2
votes
1
answer
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Diophantine equation $10^n-a^3-b^3=c^2$
Consider the Diophantine equation:
$10^n-a^3-b^3=c^2$, for $a$, $b$, $c$, and $n$ positive.
Has this equation infinitely many solutions?
16
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1
answer
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Connection between isomorphisms of algebraic topology and class field theory
I am considering the following two isomorphisms:
First, if $X$ is a reasonably nice topological space, then $X$ has a normal covering space which is maximal with respect to the property of having an ...
- 4,618
3
votes
1
answer
332
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Hilbert class field tower
Let $K$ is a number field,and $H_{K}^{i},i=1,2,\cdots$ be its Hilbert class field tower,suppose it is finite,and let $L=H_{K}^{n}$ is the top of the tower. Must $L$ be galois over $K$?
- 4,933
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1
answer
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Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?
Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{...
- 99.2k
4
votes
0
answers
235
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A connection between basic hypergeometric series and number theory
I am studying functions given by the power series:
$$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$
The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
CommunityBot
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1
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Inequality in Iwaniec-Kowalski
I am reading about Dirichlet polynomials in the book Analytic Number Theory by the said authors. Can anyone justify the following inequality? Assume that $a(n),b(m)$ are sequences of non-negative ...
CommunityBot
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1
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What's the simplest rational not expressible as a sum of a given number of unit fractions?
This is essentially the same as the closed question Representation of rational numbers as the sum of 1/k but I hope I can make a case for it as an MO-worthy question.
Ed Pegg, Jr., in his Math Games ...
10
votes
1
answer
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Iwasawa theory and perfectoid spaces
Have there been any applications of perfectoid theory to Iwasawa theory? At a first glance, this seems like a natural choice. For instance, the field $\mathbb Q_p(\mu_p^{1/p^\infty})$ is studied in ...
- 36.2k
4
votes
0
answers
361
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Kottwitz global gerbes
I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...
- 33.9k
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0
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Given a prime $\,p\ne3$, is it always possible to find another prime q such that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$?
Given a prime $\,p\ne3$, is it always possible to find another prime q such
that $\,\phi(q)=\phi(p\cdot2^{2n+1})\,$ for some $n\in\mathbb{N}$ ($\,\phi\,$ is the Euler's totient function)?
Some ...
- 1,861
4
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2
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Positions in the Wythoff array
Suppose that $x$ and $y$ are positive integers. How can the position of $x+y$ in the Wythoff array (A035513) be predicted from the positions of $x$ and $y$?
Background. The Wythoff array begins with
...
- 30k
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1
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Find a cubic monic integral coefficient whose discriminant is $-83$
I trie to calculate the Hilbert class field of $\mathbb Q\left(\sqrt{-83}\right)$, whose class number is $3$, so I should find a cubic integral monic polynomial whose discriminant is $-83$, but I ...
- 45.8k
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0
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35
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Is there any characterization of cototients equal to prime powers?
Do we know something about numbers $x$ with
$x-\varphi(x)=p^n$ for some $n$ and prime $p$
Are there any for each prime $p$?
- 333
1
vote
0
answers
95
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Efficiently computing the digits of irrational number
Is there irrational real number $C$, defined by algebraic numbers and elementary functions such that the $n$-th digit in base $b$ in the fractional part
is computable in time polynomial in $\log{n}$?
...
- 24.2k
11
votes
1
answer
421
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How many numbers $\le x$ can be factorized into three numbers which form the sides of a triangle?
Note: Posting in MO since it was unanswered in MSE
Definition: We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3, 1 \le d_1 \le d_2 \...
- 43.3k
-1
votes
1
answer
282
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Is it possible to determine whether the sequence $\,a_0=p,\;a_{n+1}=(a_n-2)\cdot a_n+2\,$ will reach another prime number?
Given a prime $\,p\,$ let's consider the following sequence:
$a_0=p$
$a_{n+1}=(a_n-2)\cdot a_n+2$
Is it possible to determine whether the sequence $\,a_n\,$ will reach, sooner or later, another ...
- 1,000
6
votes
0
answers
339
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Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
CommunityBot
- 1
4
votes
2
answers
435
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Nonvanishing of central L-values of Maass forms
Are there any results on the proportion of nonzero central L-values of Maass cusp forms? More precisely, I am looking for lower bounds for
\begin{equation} \frac{\#\{\phi_j : \, L(1/2, \phi_j) \neq 0,...
- 21
11
votes
4
answers
4k
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Variants of Eisenstein irreducibility
In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
- 60.4k
3
votes
1
answer
398
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Which Hilbert's 10th polynomials are known to have solutions?
The Diophantine equation
$$x^3 + y^3 + z^3 = 42$$
was recently solved by
Booker and Sutherland:
Sum of three cubes for 42 finally solved.
Is there a clean partition of the form of those
polynomial ...
CommunityBot
- 1
1
vote
0
answers
66
views
Approximate a given number by factorials
For a pair of integers $a$ and $b$, can we approximate a given number by $\log_{a^{!^n}}b^{!^m}$ where $a^{!^n}$ is iterated factorial operator? Are there theorems to deal with this kind of questions?
