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
14,028
questions
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34
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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?
-1
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0
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33
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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 [...
- 39
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votes
0
answers
82
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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 ...
- 151
1
vote
1
answer
124
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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 ...
- 39
1
vote
0
answers
53
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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
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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 ...
- 39
3
votes
1
answer
107
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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 ...
- 151
3
votes
3
answers
1k
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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 $A$.
...
- 23.3k
13
votes
2
answers
1k
views
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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0
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76
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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<\...
- 37
2
votes
1
answer
71
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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 $$ (2)\text{ }...
- 123
2
votes
1
answer
190
views
Euler's totient phi and a prime
Let $p$ be a prime and $n$ a positive integer not divisible by $p$. When working on the fixed field in the cyclotomic field $Q(e^{2i\pi/n})$ of the Galois automorphism $Gal(Q(e^{2i\pi/n})/Q)\ni\tau:\...
- 135
1
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0
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79
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Does the Ramanujan-Petersson condition correspond to a Fourier type property?
The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
- 6,672
3
votes
0
answers
118
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Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology
Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...
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votes
1
answer
2k
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Has this number-theoretic constant been studied?
Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)...
- 449
1
vote
0
answers
91
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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
0
answers
57
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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 ...
- 121
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votes
0
answers
129
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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
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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 ...
- 113
2
votes
0
answers
82
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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 ...
- 121
2
votes
1
answer
120
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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 ...
- 977
3
votes
0
answers
85
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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 ...
2
votes
2
answers
215
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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
$3x^2+3y^2+z^2=t^2$
...
- 21
1
vote
1
answer
135
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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:
$$x \oplus y := ((x+y) \...
- 1,233
5
votes
2
answers
551
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Specific application of Cauchy-Schwarz and Large Sieve
Im reading a paper by Matomaki here, and the following is stated (I'm paraphrasing):
"By the Cauchy-Schwarz inequality and the large sieve, we have
$$\sum_{q \leq Q}\frac{q}{\phi(q)}\sum_{\...
- 153
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?
- 12.8k
1
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0
answers
79
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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{...
- 1,489
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 $C$ ...
- 888
5
votes
1
answer
125
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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{...
- 695
2
votes
0
answers
76
views
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
124
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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\...
- 615
4
votes
1
answer
256
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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 ...
- 105
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 ...
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....
- 1,059
0
votes
0
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50
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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) =...
- 131
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 ...
- 411
1
vote
0
answers
199
views
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 ...
- 1,149
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 ...
- 159
3
votes
2
answers
286
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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 ...
- 31
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 ...
- 1,180
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 ...
- 21.3k
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 ...
- 105
1
vote
1
answer
151
views
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 ...
- 868
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 ...
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. ...
- 111
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 ...
- 376
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 ...
- 11
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 ...
- 21.3k