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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About arithmetic functions

What are the new researches about the arithmetic functions that are good and interesting in the area of Mathematics?
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Diophantine equations or associative operations on ordered lattice

I am a person in the third world country and in our universities the advisors propose the the topics to research to students, and there  two topics for research have proposed to me the first one is [...
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82 views

Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

Here is what I observed : Inspired by Lucas-Lehmer primality test, I think I made a primality test for numbers of the form $\frac{a^p-1}{a-1}$ but the test isn't perfect and there are some conditions ...
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1 vote
1 answer
124 views

Diophantine equations with arithmetical functions

I want to know is the diophantine equations that contain arithmetic functions are an interesting topic to research? (For example $\varphi(x)=cx-1$ and $\varphi(x)=\sigma(x)-1$.) $\sigma(x)$ is the sum ...
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1 vote
0 answers
53 views

Properties of universal deformation ring

Let $ \mathbb{F} $ be a finite field of characteristic $ \ell>0$ and $ W(\mathbb{F})$ its ring of Witt vectors. As usual, in the context of Galois deformation theory, let $ G $ be a profinite group ...
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3 votes
1 answer
160 views

Diophantine equations

It has been proved that there is no algorithm to solve Diophantine equations, for that reason I want to know what are the Diophantine equations that physicists or chemists need to solve? Or any other ...
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3 votes
1 answer
107 views

Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$

Here is what I observed: For $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : Let $N$ = $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : when $m$ is a number $m \ge 3$ and $n \ge 0$. Let the ...
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3 votes
3 answers
1k views

Surprisingly long closed form for simple series

For natural $A$ define $$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$ $f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$. ...
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13 votes
2 answers
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Polynomial values are powers of two

The initial question comes from Komal in 1999. Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
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2 votes
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+50

Conjectures inspired in the context of Casas-Alvero conjecture, via the logarithmic derivative of derivatives of a polynomial

In the post (cross-posted in Mathematics Stack Exchange with identificator MSE 4244256 and same title) we assume that $P(x)=a_0+a_{1}x+\ldots+a_{n-1}x^{n-1}+a_{n}x^n$ is a polynomial of degree $1<\...
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2 votes
1 answer
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A condition on $(a_{j})_{j\in \mathbb{N}}$ so that for all $x \in \mathbb{R}$ we have $\min_{1 \leq j \leq N}\|a_{j}x\|=o(1)$

Suppose that the sequence $(a_{j})_{j \in \mathbb{N}}$ is an increasing sequence of positive integers that satisfies $$(1)\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } d | a_{d}$$ and $$ (2)\text{ }...
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2 votes
1 answer
190 views

Euler's totient phi and a prime

Let $p$ be a prime and $n$ a positive integer not divisible by $p$. When working on the fixed field in the cyclotomic field $Q(e^{2i\pi/n})$ of the Galois automorphism $Gal(Q(e^{2i\pi/n})/Q)\ni\tau:\...
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Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
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3 votes
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118 views

Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology

Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
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10 votes
1 answer
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Has this number-theoretic constant been studied?

Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)...
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1 vote
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On probability of coprimality of a list of numbers

We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$. Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
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1 vote
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Forming rational numbers using unique Egyptian fractions

Question: For a given rational number $r\in (0,1)$, does there exists a finite, ordered set $S\subset \mathbb{N}$ such that the product of the first $k$ elements of $S$ do not divide the $k+1$th ...
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3 votes
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What can be said about the primality of Zsigmondy numbers?

I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months. Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
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3 votes
1 answer
297 views

Supercuspidal, spherical and discrete series representation

Let $G$ be an algebraic group defined over $\mathbb{Q}$ with maximal unipotent radical $N$. Let $\pi$ be an admissible representation of $G(\mathbb{Q}_p)$, we say that this representation is ...
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2 votes
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Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...
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2 votes
1 answer
120 views

Difference sequences of sets of integers

In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...
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3 votes
0 answers
85 views

The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$

Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
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2 votes
2 answers
215 views

Parametrization of integral solutions of $3x^2+3y^2+z^2=t^2$ and rational solutions of $3a^2+3b^2-c^2=-1$

1/ Is it known the parameterisation over $\mathbb{Q}^3$ of the solutions of $3a^2+3b^2-c^2=-1$ 2/ Is it known the parameterisation over $\mathbb{Z}^4$ of the solutions of $3x^2+3y^2+z^2=t^2$ ...
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1 vote
1 answer
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A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...
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5 votes
2 answers
551 views

Specific application of Cauchy-Schwarz and Large Sieve

Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing): "By the Cauchy-Schwarz inequality and the large sieve, we have $$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
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4 votes
1 answer
173 views

Is there an Erdős–Kac theorem for number of divisors?

Erdős–Kac theorem gives average number of prime factors of an integer. Is there a theorem which concerns average number of divisors of an integer?
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1 vote
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A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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2 votes
0 answers
69 views

Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?

Let \begin{equation*} \zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}} \end{equation*} be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation \begin{...
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5 votes
2 answers
747 views

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
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5 votes
1 answer
125 views

Describing the Gamma-transform explicitly in terms of power series

The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is: $$\Gamma_{\alpha}(s) = \int_{\mathbf{...
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2 votes
0 answers
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Computing coefficients of theta functions associated to quadratic forms

If we take an integral positive definite quadratic form $Q$ and set $\Theta_Q(z) = \sum_{k\geq 0}R_Q(k)e^{2\pi ikz}$, what are the most efficient algorithms to compute the $R_Q(k)$? I am aware e.g. of ...
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1 vote
1 answer
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Completion reducing to localization on Noetherian rings

It is quite easy to show that if $A$ is a Dedekind domain and $\mathfrak{p}\in \operatorname{Spec} A$, then if $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$ and $A_{(\mathfrak{p})}=(A\...
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4 votes
1 answer
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Rationality of field embeddings

After my earlier question question turned out to have a negative answer (Thank you to all respondents!), here is a more modest one. Both a positive answer and a counterexample would help my work. If ...
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1 vote
1 answer
290 views

Square root of prime numbers

I found that the square root of any prime number S can be approximated, at the n-th order, as a rational number represented by the polynomials shown below. $x_0$ is an initial seed, which is a ...
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4 votes
0 answers
107 views

Primes of supersingular reduction for non-CM elliptic curves

When $E/\mathbb{Q}$ is a non-CM elliptic curve, Serre had shown that there are density 0 primes of supersingular reduction. His proof can be generalized to elliptic curves over arbitrary number fields....
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Decidability of a polynomial-exponential equation in two variables

My question is with regards to the following (algorithmic) problem: Problem. Given $f\in \mathbb{Z}[x,y], a,b\in \mathbb{Q}, r\in \mathbb{Z}$, do there exist positive integers $m,n$ such that $f(m,n) =...
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1 vote
0 answers
160 views

Prime numbers formed by consecutive numbers

Let $x$ be an even number. Suppose that we concatenate it with its successor to form $x (x+1)$ (not multiplication, but concatenation). For example, if we start with $x = 2$, we would get $23$. If we ...
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1 vote
0 answers
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Advanced texts on analytic number theory?

So a friend of mine is very interested in analytic number theory, and is looking for resources past the basic level. He has studied analytic number theory from several books, among them are Hardy’s ...
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2 votes
2 answers
537 views

What is the most "informative" Yes/No math question you know? [closed]

Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a ...
7 votes
0 answers
128 views

Finding when a certain product in a cyclotomic field is equal to one

For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
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3 votes
2 answers
286 views

Question about iterations not divisible by infinitely many prime numbers

For a given $a\in \mathbb{Z}$, define $P(a)$ to be the set of all prime numbers dividing $a$. Also define $\mathcal{P}$ to be the set of all prime numbers. Let $a,b,c\in \mathbb{Z}\setminus \{0\}$ be ...
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1 vote
0 answers
80 views

Image of the Kummer map for abelian varieties over $p$-adic local fields

The following statement might be well-known to the community: let $K$ be a finite extension of $\mathbb{Q}_p$ for some prime $p$. Let $A$ be an abelian variety over $K$. Then the image of the Kummer ...
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0 votes
0 answers
60 views

Expectation of edge weights on the complete graph, Part 2

This question concerns the same basic set-up as my previous question: Expectation of edge weights on the complete graph In that question an answer was given which shows that the expected value is as ...
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6 votes
2 answers
253 views

Cancellation of irreducibility for Galois conjugates

Motivation: Take an algebraic number $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field ...
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  • 105
1 vote
1 answer
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Non-negativity of an infinite absolutely convergent sum

The infinite sums involving mobius function and a multiplicative function has got quite interest in past. In particular, sums of the form $$\sum_{d=1}^{\infty}\frac{\mu(d)}{f(d)}$$ for mobius function ...
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4 votes
1 answer
648 views

Is there a way to specify a special kind of reciprocals of natural numbers?

Any number with of a form $\frac{1}{n}$ has a decimal with a repetend of finite length that is never longer than $n$ (provable by Dirichlet principle). (Example: $\frac{92}{99}=0.929292\ldots$ in ...
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7 votes
0 answers
232 views

Analogs of the Weil conjectures for non-archimedian fields

Suppose that $X$ is a smooth and proper variety defined over a perfect non-archimedian valued field $k$ of characteristic $p$.  Then one can consider the action of Frobenius on crystalline cohomology. ...
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0 votes
1 answer
137 views

Asymptotic behavior of the sum $\sum_{k\le x}\frac{1}{\varphi(k)}$

Suppose $x>0$ and let $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic ...
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1 vote
0 answers
62 views

Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
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1 vote
1 answer
131 views

Expectation of edge weights on the complete graph

Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
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