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
15,907
questions
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Reference request - logarithmic average
Consider a set $A\subseteq\mathbb{N}$. Consider an arithmetic function $a(n):\mathbb{N}\to\mathbb{C}$. I am looking for notation which describes the following:\begin{equation}\frac{\sum_{n\in A}\frac{...
26
votes
1
answer
2k
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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 ...
2
votes
0
answers
80
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Limit involving the fractional part and the Fibonacci numbers
Helo,
Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving
$$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\...
57
votes
5
answers
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Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture
Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635
In that preprint, Kirti Joshi claims that
he agrees with Scholze and ...
11
votes
2
answers
264
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Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?
Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.
In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in ...
0
votes
0
answers
123
views
On hashing prime numbers into prime number of buckets
Let $b$ be any prime. Consider a set of $b-1$ buckets. Consider all prime numbers (except $b$) up to some $N$. Let us do the simple hash wherein each prime $x$ less than $N$ is assigned to the $x \...
1
vote
1
answer
91
views
Refinement of a theorem of Koblitz-Ogus
In their appendix to Deligne's paper "Valeurs de fonctions L et périodes d'intégrales" (PSPM 33, 1979), Koblitz and Ogus prove that functions $N^{-1}\mathbf{Z}/\mathbf{Z}-\{0\}\to \mathbf{Q}$...
16
votes
2
answers
1k
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Is it decidable whether two real algebraic irrationals generate the same extension of the rationals?
For an algebraic number $\alpha$, let $f_\alpha$ denote its minimal polynomial. We can symbolically represent an algebraic number $\alpha$ by the tuple
$$
(f_\alpha, x, y, r) \in \mathbb{Q}[x] \times ...
4
votes
1
answer
168
views
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 ...
7
votes
2
answers
2k
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Is there any progress toward solving Gilbreath's conjecture?
Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the ...
4
votes
1
answer
182
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Do Artin L functions have polynomial growth in the critical strip?
Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
3
votes
1
answer
601
views
Geometric mean of prime factors of all numbers up to n
Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
1
vote
0
answers
193
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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 $...
2
votes
1
answer
379
views
Chinese remainder theorem for target interval
Given $n$ pairwise coprime natural numbers $m_{1}, \dots, m_{n}$ with remainders $y_{i}$, for all $i \leq n$. Furthermore, we have a target interval $I := \left[ a, b \right]$, with $1 \leq a < b \...
10
votes
2
answers
778
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Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?
In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
-1
votes
0
answers
51
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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, ...
1
vote
0
answers
119
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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 $\dim_{\mathbb{Q}_p} V$...
0
votes
0
answers
47
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 ...
1
vote
1
answer
167
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Structural differences between closed forms of two related infinite products?
In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way.
Take $a \in \mathbb{R}, s \in \mathbb{C}$ and:
$$\displaystyle C(s,a) := \prod_{n=1}^\infty \...
2
votes
2
answers
714
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Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all non zero Integers.
How to find All solutions ?
Is there any parametrization which gives Infinitely ...
14
votes
4
answers
3k
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Does Weyl's Inequality prove equidistribution?
Let $f(n) = \theta n^d + a_{d-1} n^{d-1} + \cdots a_1 n + a_0$ be a polynomial with real coefficients, and $\theta$ irrational. Let $S_N = \sum_{n=1}^N e^{2 \pi i f(n)}$. Weyl's Equidistribution ...
3
votes
1
answer
150
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Kernel of a map of Tate algebras
Let $A$ and $B$ be a pair of noetherian Tate algebras over $\mathbb{Q}_p$, and assume $\text{dim}_{\text{Krull}}(B) > \text{dim}_{\text{Krull}}(A)$. Is it true that any morphism $B \longrightarrow ...
11
votes
3
answers
817
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Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$
I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
0
votes
1
answer
88
views
Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
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 ...
4
votes
0
answers
106
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 $\...
1
vote
0
answers
72
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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 ...
7
votes
1
answer
365
views
Integral refinements of rigid cohomology
Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
9
votes
1
answer
2k
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What keeps asymptotic Goldbach's conjecture out of reach of current technology?
Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
3
votes
0
answers
75
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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-...
0
votes
0
answers
143
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 ...
8
votes
3
answers
580
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Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
It seems numerically up to $n \leq 10^6$ that for $m=3$ or $...
27
votes
1
answer
1k
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About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I ...
3
votes
0
answers
314
views
Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?
I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
11
votes
4
answers
1k
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Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
5
votes
0
answers
135
views
A puzzle with magic Egyptian tilings
Background
I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
0
votes
0
answers
34
views
Permutation for the fast computing of the $k$-th partition of $n$ in reverse lexicographic order
Let $a(n)$ be A000041 (i.e., $a(n)$ is the number of partitions of $n$ (the partition numbers)).
Let $b(n)$ be A000070. Here
$$
b(n) = \sum\limits_{i=0}^{n}a(i)
$$
Let $c(n)$ be $k-1$ where $k$ is the ...
11
votes
2
answers
497
views
Jacobi symbols for two-square sums of primes
Given a prime $p\equiv 1\pmod 4$, Fermat's two-squares theorem discovered by Girard
states that there exists two integers $A,B$ such that
$p=A^2+B^2$.
For all primes up to $10^7$ the integers $A$ and $...
1
vote
2
answers
180
views
Direct algorithm for an integer program
Let $p$ be a prime and let $h_1,h_2\in\{1,2,\dots,p-1\}$ be integers.
Is there any direct algorithm to solve for following in polynomial in $\log p$ time?
$$\min (x_1-x_2)^2$$
$$x_1,x_2,k\in\mathbb Z$$...
3
votes
1
answer
277
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 ...
2
votes
0
answers
105
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 ...
-1
votes
0
answers
54
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
votes
1
answer
144
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 ...
4
votes
1
answer
156
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 ...
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$. ...
5
votes
1
answer
225
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 ...
4
votes
1
answer
149
views
Meaning of dagger cohomology $H^{1 \dagger}(G^\dagger)$ in "Frobenius and Monodromy Operators" by Coleman and Iovita
Let $G$ be an abelian variety with good reduction over a finite extension $K$ of $\mathbb{Q}_p$. In equation (2.4) on page 179 of my edition of "The Frobenius and monodromy operators for curves ...
0
votes
0
answers
101
views
Unimodality of the Stirling numbers
For fix $n$, the (unsigned) Stirling number of the first kind $c(n,k)$ and the Stirling number of the second kind $S(n,k)$ are both unimodal.
Erdős Paul proved the sequence $c(n,k)$ has a unique mode ...
30
votes
4
answers
3k
views
Motivation for zeta function of an algebraic variety
If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}_p$ is defined to be
$$\zeta_{V,p}(s) := \exp\left(\sum_{m\geq 1} \frac{N_m}{m}(p^{-s})^m\right). $$
where $N_m$ is ...
5
votes
1
answer
414
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 ...
-1
votes
0
answers
42
views
Find transseries from difference equation [closed]
I want to find a method to solve equations of the form
$f(x+1)=f(x)+g(x)$ for a given function $g$ and $f(x)=0$.
The paper here has solutions for $f(x+1)=\lambda(x)f(x)+g(x)$, which is more general ...