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
15,947
questions
4
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1
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Conditional convergence of Artin $L$-functions
Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
- 20.8k
1
vote
2
answers
167
views
Refinement of a theorem of Koblitz-Ogus
In their appendix to Deligne's paper "Valeurs de fonctions L et périodes d'intégrales" (PSPM 33, 1979), Koblitz and Ogus prove that functions $N^{-1}\mathbf{Z}/\mathbf{Z}-\{0\}\to \mathbf{Q}$...
- 502
16
votes
2
answers
1k
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Is it decidable whether two real algebraic irrationals generate the same extension of the rationals?
For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...
- 1,025
2
votes
0
answers
141
views
Bounding dimensions of Galois cohomology
Let $G$ be the absolute Galois group of the rationals, and $V$ a finite dimensional $p$-adic representation.
Is there a way to provide a general bound on $H^i(G, V)$ in terms of $\dim_{\mathbb{Q}_p} V$...
- 1,641
0
votes
0
answers
48
views
Constructing squares using linear operations when a sizeable residue is given
Given $x\in\{0,1,\dots,2^k-1\}$ and given $x^2\bmod p$ where $p$ is a prime at in $[2^k,2^{k+1}]$ is it possible to construct $x^2$ using only at most $O(2^{k})$ linear in $x$ operations (that is ...
- 13.7k
4
votes
0
answers
113
views
Do all nonnegative integers appear in A051521?
For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$，they form a rational sequence
1,1,3/2,4/3,5/2,3/2,…
Because $\...
- 311
-2
votes
1
answer
102
views
Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)),
$$
where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$.
I want ...
- 363
3
votes
0
answers
95
views
The criterion for dimensional conjecture for universal Galois deformation rings
I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
- 817
60
votes
4
answers
36k
views
Global character of ABC/Szpiro inequalities
In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro ...
- 737
0
votes
0
answers
159
views
Why $k((x,t))$ can not be a local field?
If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field.
I ...
- 870
5
votes
0
answers
148
views
A puzzle with magic Egyptian tilings
Background
I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
- 4,575
2
votes
0
answers
258
views
On $(k,\ell)$-sumfree sets
Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation
$$x_1+\dots +x_k = y_1+\dots +y_\ell$$
in the set (for distinct $x_i$'s and $...
- 781
-1
votes
0
answers
55
views
Continuous version of ergodic with integral
Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
- 313
11
votes
4
answers
1k
views
Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
- 311
1
vote
0
answers
73
views
In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?
Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements:
$\lambda$ being a random large prime such as $w^\lambda > 2\times m$
$1 < n < m−1$.
m is ...
- 151
-1
votes
0
answers
57
views
Does algebraic independence of logarithms conjecture imply L-W?
Assume that algebraic independence of logarithms conjecture is true: Let $a_1,\cdots,a_n$ be algebraic numbers such that $\log(a_1),\cdots,\log(a_n)$ are $\mathbb Q$-linearly independant. Then, $\log(...
- 3,739
3
votes
1
answer
288
views
Unique factorization of ideals in a quadratic field
"Suppose $k = \mathbb{Q}(\sqrt{d})$ is a real quadratic field ($d > 1$ a square-free integer) with fundamental unit $\varepsilon$, normalized as usual so that $\varepsilon > 1$ with respect ...
3
votes
0
answers
355
views
Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?
I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
- 2,462
1
vote
2
answers
185
views
Direct algorithm for an integer program
Let $p$ be a prime and let $h_1,h_2\in\{1,2,\dots,p-1\}$ be integers.
Is there any direct algorithm to solve for following in polynomial in $\log p$ time?
$$\min (x_1-x_2)^2$$
$$x_1,x_2,k\in\mathbb Z$$...
- 13.7k
4
votes
1
answer
164
views
Characters on ray class groups
Let $K$ be an algebraic number field, $\mathcal{O}_K$ its ring of integers, $\mathfrak{m}$ an integral ideal of $\mathcal{O}_K$. Let $J$ be the set of all fractional ideals, $P$ the set of principal ...
- 1,888
5
votes
1
answer
234
views
Converse of "generalized Hilbert 90" / Galois descent
The following generalization of Hilbert 90 can be found in Serre's Corps Locaux (Chap. X, §1, ex.2, p.160 of the French edition), see also this question:
Theorem: If $L|K$ is a finite Galois extension ...
- 313
3
votes
1
answer
153
views
Triangular repdigits
I would like to know whether $55$, $66$ and $666$ are the only triangular numbers that are repdigits, i.e., numbers at least $10$ whose digits w.r.t. base 10 all agree.
In more sophisticated terms, I ...
- 11.9k
0
votes
0
answers
136
views
On hashing prime numbers into prime number of buckets
Let $b$ be any prime. Consider a set of $b-1$ buckets. Consider all prime numbers (except $b$) up to some $N$. Let us do the simple hash wherein each prime $x$ less than $N$ is assigned to the $x \...
- 5,493
6
votes
1
answer
437
views
Nice diophantine equations with large smallest solutions
Given a polynomial $P$ with integer coefficients in finitely many variables,
we denote by $v(P)$ the product of the absolute values of the non-zero coefficients
and the non-zero total degrees of the ...
20
votes
1
answer
1k
views
Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$
This question is first asked by me on MSE, but I haven't recieve a nice answer yet.
I would like to determine whether the polynomial $p(x)=x^n+5x+3$ is irreducible over $\mathbb{Q}$ when $n\ge 2$. ...
- 315
1
vote
0
answers
58
views
If we allow DH operations in addition to exponentiation and multiplication can we get a lower bound for discrete logarithm?
In https://crypto.stackexchange.com/questions/72969/proof-dlog-is-hard-in-generic-group-model/ it is shown if we allow only exponentiation and multiplication we can get an exponential complexity lower ...
- 13.7k
25
votes
2
answers
858
views
Proofs of the valence formula that avoid tricky contours?
$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...
- 109k
12
votes
2
answers
1k
views
Prime differences and zero multiplicity
Concerning gaps between consecutive primes, Paul Erdős conjectured that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
- 232
2
votes
0
answers
90
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
- 13.6k
4
votes
1
answer
491
views
Subgroup of p-adic units
Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
- 119
2
votes
0
answers
112
views
A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
- 41
2
votes
0
answers
101
views
Gaussian primes in translations of lattices in $\mathbb{Z}[i]$
I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} ...
- 21
3
votes
0
answers
163
views
Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
- 11
7
votes
1
answer
534
views
Original proof of Hilbert irreducibility theorem
Does there exist a modern exposition of Hilbert's original (1892) proof of the Hilbert irreducibility theorem? Of course, I can (and will) read Hilbert's original article, but I would feel more ...
- 1,130
1
vote
1
answer
217
views
Existence of odd mod $p$ Galois representations whose image is $p'$-group
Let $K$ be a number field and let $G_K$ be the absolute Galois group of $K$. Let $p$ be an odd prime and $\mathbb{F}_p$ be a finite field of order $p$. Can we always find a continuous representation $\...
- 817
1
vote
0
answers
114
views
Multiplicities of Galois representations in the semisimplification of the reduction of a Tate module
Let $C$ be a smooth proper curve, of genus $g$ over a number field $K$. Let $v$ be a prime of good reduction for $C$ above $p>2$, and let $T_pJ$ denote the $p$-adic Tate module of $J$, the Jacobian ...
- 1,641
0
votes
0
answers
136
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A question and reference about Bombieri's article continued fraction of algebraic numbers
Above the Comments in the article continued fraction of algebraic numbers, there are some words on the unboundedness/cycle of coefficients of continued fraction of algebraic numbers "Thus, ...
- 3,000
2
votes
0
answers
65
views
When is the number-theoretic transform of small vectors again small?
I am currently working on an idea in the context of lattice-based cryptography, but the problem that I am currently stuck on seems to have almost nothing to do with lattices anymore.
In particular, my ...
- 121
3
votes
0
answers
424
views
Analytic number theory and condensed mathematics
As of 2024, are there current or planned applications of condensed mathematics to analytic number theory? If so, what are suggested readings?
- 737
-1
votes
0
answers
70
views
On the full list of near-repdigit perfect powers
I'm interested in the full list of perfect powers ($a^b$ where $a, b \in \mathbb{Z}$, $a \ge 1$, $b \ge 2$) that are near-repdigit in base 10. A near-repdigit is a $k$-digit number where $k \ge 2$ and ...
- 99
2
votes
0
answers
56
views
How to check that a number probably/likely has a divisor having a specific bit length/in range?
Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
- 151
2
votes
0
answers
85
views
Is the extension field by zeros of $x^{2m}-p^{2(m-1)}=0$ over $\mathbb Q_p$ totally ramified?
Let $m \geq 2$ be an integer. Consider the polynomial $f(x)=x^{2m}-p^{2(m-1)} \in \mathbb{Q}_p[x]$.
I want to study the field extension by the zeros of $f(x)$ over $\mathbb{Q}_p$.
What is the degree ...
- 870
9
votes
0
answers
162
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
- 40.9k
2
votes
2
answers
743
views
Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all non zero Integers.
How to find All solutions ?
Is there any parametrization which gives Infinitely ...
- 47
2
votes
0
answers
107
views
How to know if a random natural number is a probable semiprime?
Let that $n\in\Bbb N$ generated from a hash function where $n$ is long enough to be hard to factor in the gnfs algorithm. How to check if $n$ is probably a semi‑prime in a faster way than factoring it ...
- 151
3
votes
0
answers
191
views
Do all polynomials (other than generalized cyclotomic polynomials) have the spaced polynomial property?
Anna Erschler just asked me a question that is posed as Question 1.2 in her recent preprint with J. Frisch and M. Rychnovsky. I am asking it here with her permission - since I find it interesting (...
- 19.4k
1
vote
0
answers
55
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
5
votes
0
answers
381
views
How to prove these identities for $\log(2)$ based on $_3F_2$ integrals?
In this MO post I have placed 4 Ramanujan-type hypergeometric series found using the LLL algorithm for fast computing of some logarithms. I could prove 3 of them by means of classical methods based on ...
- 2,210
1
vote
0
answers
117
views
A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
- 474
18
votes
1
answer
3k
views
Assuming the Collatz conjecture is false, what is known about the size of the false set?
If the Collatz conjecture is strongly false, in the sense that there is an infinite orbit, let $S_n$ be the set of natural numbers $\le n$ whose orbit goes to infinity.
If $c=\liminf _{n\rightarrow\...
- 5,045