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

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11 votes
2 answers
293 views

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
64 views

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
3k 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 ...
2 votes
0 answers
80 views

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
37k views

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 ...
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
92 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 views

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 views

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 views

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
194 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 $...
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 views

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 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, ...
1 vote
0 answers
119 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 $\dim_{\mathbb{Q}_p} V$...
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 views

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
715 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 ...
14 votes
4 answers
3k views

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 views

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 views

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 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 ...
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 views

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
76 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-...
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 views

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 views

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
315 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 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 ...
5 votes
0 answers
136 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 ...

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