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
15,592
questions
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Reference book between Riemann z function and random matrix
What is a reference book to understand the relation between Riemann z function and random Matrix
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Invariant factors and commuting matrices over a discrete valuation ring
Let A be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T, Y^T$ be their transpose. Assume that $$p^dA^n\subset Im X+Im Y.$$
Does ...
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17
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Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
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vote
1
answer
82
views
Quadratic unramified extension of a p-adic field
Let $F$ be a non-archimedean local field of residual characteristic $p\neq 2$, and let $E=F[\sqrt{\epsilon}]$ be the quadractic unramified extension, here $\epsilon$ is a non-square element of $\...
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1
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1
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A Mordell equation $y^3=x^2+20$ [closed]
Recently I met a problem when I'm studying algebraic number theory.
Problem. Find all positive integer solutions of $y^3=x^2+20$.
I solved the situation when $x$ is an odd because the two ideals $(x+...
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0
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93
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Roth's Theorem Variations?
One can motivate Roth's theorem as follows. On $[0,1]$ consider the function $f$ that takes
$x$ to the cardinality of the set
{ $p/q : |x - p/q | < 1/q^{2 + \epsilon}$ } .
Now one can see that $\...
- 17.2k
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3
answers
265
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Closed formula for number of ones in a proper factor tree
Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...
- 7,624
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93
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Reference request: unfolding of Integral representation of an L-function
Are there any text or papers that thoroughly address unfolding of integral representation of an L-function such as D. Ginzburg's On Spin L-function for Orthogonal Groups page 762-763 and page 774 (or ...
- 111
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1
answer
177
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
- 4,783
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75
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How to write down the contrapositive of a statement and prove if it's right or not [closed]
I have been battling with my professor over this question for months. Every time I come
up with an answer she tells me wrong.
Question:
Consider the following proposition concerning an integer n ≥ 2. ...
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3
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2
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180
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Integer solutions to $x^2 + x + 1 = y^z$?
In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on ...
- 1,153
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1
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98
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On a summation in "Artin's conjecture for primitive roots" by Heath-Brown
This is a question on the paper: D. R. Heath-Brown. Artin's conjecture for primitive roots. Quarterly J. Math. 37 (1): 27–38, 1986.
At the beginning of the proof of his main theorem on page 35, Heath-...
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Is it possible to use the Cauchy-Schwarz Inequality on the following sum : $\sum_{j=0}^{\infty}\frac{(-1)^{j}(k-\frac{1}{3})^{j+k}}{j!(j+k)}$
I'm working on a small paper that involves this sum, and I was wondering if you could separate the components of the sum and square them, using the Cauchy-Schwarz Inequality.
The paper is centered on ...
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answers
60
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Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,708
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283
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Cubic twist of elliptic curves and its rank
Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$).
Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$.
$E$ and $E_D$ are isomorphic over $\...
- 1,325
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1
answer
226
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Radicands of square roots of the 2020s, written in simplest radical form
As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
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241
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Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?
Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...
- 4,783
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1
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267
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How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
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Prove that this equation for natural m and n doesnt have an answer [closed]
$19^(19)=m^3 + n^4$
from $19^(19)$ i mean 19 to the power of 19
i've tried m and n for mod k, k=1,2,...,11 but i haven't reached a solution
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Power summing function [closed]
f(x,p)=sum(n=1,n<=x,n^p) where p and x are integers. f(x,1)=(x^2+x)/2, and f(x,2)=x(x+1)(2x+1)/6, but what is f(x,p), where p is VERY BIG?
13
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1
answer
594
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
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0
answers
168
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Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?
In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...
- 4,783
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0
answers
24
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Enumeration of flat integral $K_4$
Question:
What is known about the enumeration of all $(a,b,c,d,e,f)\in\mathbb{N}^6_+: \\ \quad\operatorname{GCD}(a,b,c,d,e,f)=1\ \\ \land\ \exists \lbrace x_1,x_2,x_3,x_4\rbrace\subset\mathbb{E}^2:\ \...
- 12.4k
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132
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Large sets of nearly orthogonal integer vectors
This question is motivated by the Question 5 from the 2017 Asia Pacific Mathematical Olympiad. To paraphrase, the question asks what is the largest cardinality of a set $S \subset \mathbb{Z}^n$ such ...
- 24.9k
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178
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A limit involving the largest prime factor of a prime gap
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
- 4,596
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323
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+50
Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 2,193
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96
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Collision among products of binomial coefficients
Let $k, \ell \geq 2$ be positive integers. Do there exist infinitely many positive integers $N$ such that the equation
$$\displaystyle \binom{m}{k} \binom{n}{\ell} = N$$
have more than four solutions? ...
- 24.9k
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0
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92
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Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$
This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
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votes
1
answer
309
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Cohomology of Shimura varieties before and after completion at some prime
Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
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Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f,g,h,k,l~(>1)$ are natural numbers? [closed]
Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f, g, h, k, l$ are natural numbers greater than 1? The problem is related to some famous ...
1
vote
0
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66
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Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves
In this article in Compositio (2011), Munshi proves a mean value result for
$$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$
where here $f$ is a primitive holomorphic form of level $q$ with ...
- 1,061
5
votes
0
answers
116
views
Linear equation among divisors of a positive integer
Let $a,b,c$ be co-prime positive integers and let $\theta \in (1/4, 1/3)$ be a real number. For each positive integer $k$, does there exist a positive integer $N$ such that the linear diophantine ...
- 24.9k
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42
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Framed bordism and string bordism in 3-dimensions vs topological modular form
In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
- 10.3k
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1
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560
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Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in ...
- 2,112
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votes
0
answers
122
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detecting zeros of Dirichlet $L$-functions
I have been reading papers on zeros of Dirichlet $L$-functions for some of my work and I have a question. Just like the zero detecting method pointed out by Littlewood in his 1924 paper On the zeros ...
5
votes
1
answer
341
views
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
0
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0
answers
90
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Rank-24 Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant? [migrated]
A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
...
- 311
29
votes
1
answer
635
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Is there a regular pentagon with a rational point on each edge?
This question was asked by Yaakov Baruch in the comments to the question Can a regular icosahedron contain a rational point on each face? It seems that this question deserves special attention.
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1
answer
122
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Norm 1 elements of an unramified quadratic extension of a local field
Let $E$ be an unramified quadratic extension of a local field $F$, with $p$ odd. Let $E^1$ denote the set of norm $1$ elements of $E$. What can be said about the following index:
$$
{\rm 1.}\ \ \ \ [ ...
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0
answers
147
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Connection between number theory and operator theory
I was wondering if there is any connection between number theory and operator theory.
Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For ...
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1
vote
1
answer
234
views
A problem similar to the $3x+1$-problem [closed]
Let $n$ be a fixed positive integer. Define the function $f_n(x)$ as follows:
$$f_n(x)=\left\{\begin{aligned}&2x-1,\quad x\leq n;\\&2(x-n),\quad x> n.\end{aligned}\right.$$
and for $l\in\...
- 101
4
votes
0
answers
107
views
A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
- 13.4k
1
vote
0
answers
80
views
Relationship between two types of partition functions
Referring to this unanswered question on MS, I'm posting the same question here:
For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}...
- 261
-1
votes
0
answers
155
views
Size and type of a Galois radius: are they correlated?
Say as usual that a non negative integer $r$ is a Galois radius of $n$ of type $(a,b)$ if $(n-r,n+r)=(p^a,q^b)$ for some couple of primes $(p,q)$. As primes tend to get sparser as we progress along ...
- 6,954
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votes
1
answer
515
views
Central limit theorem for irrational rotations
Let $\alpha$ be an algebraic integer of modulus 1, and $ R_\alpha z=\alpha z$. Is
$$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$?
Birkhoff's ergodic ...
- 2,075
2
votes
1
answer
180
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
7
votes
1
answer
164
views
Lifting SL2(k) to a subgroup of Witt vectors
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$).
Does there ...
- 35.7k
0
votes
1
answer
179
views
Implications for large sums of roots of unity
I have some coefficients $(a_n)_{n \leq N} \subset \mathbb{R}$ such that $a_n \geq 0$ and their average value is one, i.e. $\frac{1}{N} \sum_{n \leq N} a_n = 1$. Suppose that
$$ \Bigl| \sum_{n \leq N} ...
4
votes
1
answer
285
views
Automorphic representation of GL(1)
These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.
I am reading automorphic forms from this book.
What I have understood till now:
...
- 454