Questions tagged [number-theory]
Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.
9,070
questions with no upvoted or accepted answers
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Significance Of Small Sets With Arbitrarily Long Progressions?
Does the existence of a small set with arbitrarily long arithmetic progressions (ALPs) imply that there exist large sets without any progressions at all? We can construct an example of a small set ...
- 21
2
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0
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112
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Strange Bijection in $\mathbb{Z}/(2^n-1)\mathbb{Z}$
Let $n$ be a positive integer. Find all sets $\{r_1,\cdots,r_n\} \subset \mathbb{Z}/(2^n-1)\mathbb{Z}$ such that $$\left\{ \sum_{j\in S} r_j \mid S\subset \{1,\cdots,n\}, S\ne \emptyset \right\} $$ ...
- 713
2
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109
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$\mathrm{Zi}_n(x)$ limits to $\pi(x).$ But exactly?
Consider the prime counting function: $$\mathrm{Zi}(x)=\frac{1}{e}\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\phi(k)}$$
where $$\phi(k)=\sum_{n=1}^\infty e^{-{n^k}}$$
This page shows how to sum the ...
- 1,078
2
votes
0
answers
66
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On the divergence of the series $\sum_{n=0}^{\infty }\frac{1}{1+n^{2}\cdot cos^{2}(n)}$
On the forum quora.com I encountered the question how it can be shown that this series diverges which motivated me to investigate whether there is a $c\in \mathbb{R}$ such that $n\cdot \left| cos(n) \...
- 359
2
votes
0
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47
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An inequality regarding the ceiling functions
Let $\displaystyle w(j,h)=2\sum_{k=1}^{j-1}(-1)^{k+1}\left\lceil\frac{kh}{j}\right\rceil+h-1$, prove the following conjecture: $\displaystyle \sum_{j=1\ odd}^{h-1}w(j,h)>0$.
For the alternating sum ...
- 558
2
votes
0
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50
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Example of rational jumps in upper ramification filtration
Can anyone point to me or write down an explicit example of a non-integer jump of the higher ramification filtration in positive characteristic and the corresponding equations of the intermediate ...
- 1,137
2
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54
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What does the characteristic ideal of a f.g. torsion $\Lambda(G)$-module tell me about the arithmetic of the extension?
I am currently trying to learn Iwasawa theory and am following J. Coates and R. Sujatha's book 'Cyclotomic Fields and Zeta Values'.
The setup is the following:
Let $\mathcal{F}_n:=\mathbb{Q}(\mu_{p^{...
- 45
2
votes
0
answers
35
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Linear extension of a divisors set
For a number $N$, let $S_N$ be its set of divisors, and let $C(N)$ be the number of arrangements of $S_N$ in which every divisor itself appears after all of its divisors.
$C(12)=5$, because of the ...
- 1,002
2
votes
0
answers
210
views
Non-rigorous prediction in maths through unrealistic models
Cramer's random model of the primes allows the prediction of features of the distribution of the primes even though the model is not about "real" primes but about randomly generated "...
- 29
2
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78
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For which $n \ge 2$ is $2^{1/n}$ a sum of roots of unity?
Find all integers $n\ge 2$ such that $2^{1/n}$ is a sum of roots of unity.
Source: Problem $2.18$
It seems the problem refers to finite sums (infinite sums will not converge absolutely in this case) ...
- 13.7k
2
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89
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The miraculous fish problem: sums of cubes of digits halts to 153 for multiples of 3
The fun name is a nod to the verse:
So Simon Peter climbed back into the boat and dragged the net ashore. It was full of large fish, $153$, but even with so many the net was not torn. - John 21:11
...
- 1,209
2
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When is $r^n+p\mathbb{Z}=(r+p\mathbb{Z})^n$?
Given $r, p \in \mathbb{Z}$ and $n\in \mathbb{Z}_{\ge 1}$.
Set $r^n+p\mathbb{Z}:=\{r^n+pm:m\in \mathbb{Z}\}$ and $(r+p\mathbb{Z})^n=\{\prod_{i=1}^n(r+pm_i): m_1,\dots, m_n\in \mathbb{Z}\}$.
Clearly, $(...
- 659
2
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81
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PRP testing algorithms
INTRODUCTION: (scroll down for the actual question)
I want to check the primality of large numbers (more than $100\ 000$ digits).
Moreover, I want to be able to eliminate the use of expensive (yes, in ...
- 2,158
2
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0
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57
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Maximizing a Sum over Partitions of an Interval: An Optimization Problem
Suppose $N$ is a fixed natural number. We split the interval $[1,N]$ into $L-1$ parts, where $L$ is some natural number less than $N$. This gives us an ordered set of points $\{n_i\}_{i=1}^L$, where $...
2
votes
1
answer
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Finding rational points on a circle such that $X^2+Y^2=r^2=k \in \mathbb{Z}$
I am interested in finding rational points on a circle with radius $r$, such that $r^2=k$ is an arbitrary integer. I tried reducing the problem to the unit circle, and maybe use pythagorean triples as ...
- 1,002
2
votes
0
answers
147
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Find maximum value of $y$ for which $y^2xz^3 + yx^2$ is a perfect square
Problem:
Given $y,x,z$ are positive integer variables and $N$ is a given integer constant and $x < z \le N $ and $y$ is square-free and $y \ne x$,
find the maximum value of $y$ (in terms of $x$ or $...
- 470
2
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0
answers
102
views
Does this summation span all natural numbers?
Does the range of $L(r)=1+\sum^\infty_{b=1}\sum^{b}_{a=0}(2\lfloor\frac{r}{\sqrt{a^2+b^2}}\rfloor)$ span all natural numbers, where $a, b$ are coprime integers and r is any real number greater than 0?
...
- 315
2
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56
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About "quasi-diagonal" symmetric integer matrices and their inverses
I have a symmetric, $\mathbb Z$-valued, invertible, quasi-diagonal (defined below*) $N\times N$ matrix $K$. There are two properties $K$ could have:
(a) $K$ can be turned into a block-diagonal form by ...
- 618
2
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answers
57
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Product Formula for Real Cyclotomic Polynomials
Let $n$ be a natural number, $\zeta_{n}$ be a primitive $n^{th}$ root of unity and $\Phi_{n}(x)$ be the $n^{th}$ cyclotomic polynomial. Let $\Psi_{n}(x)$ be the $n^{th}$ real cyclotomic polynomial (...
- 1,625
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0
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59
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Are there any results in the literature involving square-free factorizations of the integers?
For positive integers $n\in\mathbb{N}^*$, the radical of $n$ is defined as $\text{rad}(n):=\prod_{p\vert n}p$ where the product extends over the prime divisors of $n$. It can also be thought of as the ...
- 1,760
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0
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47
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Can I find any value super close to some value with combination of two numbers?
I hope this is not a silly question.
Suppose there is a collection of {$x^{a}y^{b}$}$_{(a,b)}$ where $x\in(0,1)$, $y>1$, and $a$ and $b$ can be any natural numbers.
I was wondering if I can always ...
2
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0
answers
50
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Behavior at cusps of two modular forms
I am a beginner, in the theory of modular forms. I came across this identity:
$\dfrac{\eta(4\tau)^6\eta(8\tau)^4}{\eta(2\tau)^4}=\displaystyle\sum_{n\geq 1}\left(\sum_{d\mid n}\chi_{-4}(n/d)\;d^2\...
- 21
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answers
46
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Prime Number Theorem: Intuition why $\alpha \pi(x_n) \sim n = \pi(p_n)$ for some $\alpha > 0$ implies $x_n \sim \frac{n \log n}{\alpha}$
I was wondering if someone could explain the intuition behind what appears to be a straightforward consequence of the Prime Number Theorem. Let $(x_k)_{k\geq 1}$ be a nondecreasing sequence in $ \...
- 567
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76
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How do I prove that this equation of arbitrarily many variables is true?
Prove that for every positive and odd integer "x" there exists a unique set of positive integers ${k_1, k_2, ... ,k_I}$ and positive integer "I", such that the following equation ...
user1146575
2
votes
0
answers
42
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First few coefficients of $\zeta_p$ as an element of an Iwasawa algebra
One way of introducing the $p$-adic Riemmann zeta function is to first define a $p$-adic pseudomeasure $\zeta_p$ via interpolation. Namely, $\zeta_p$ is uniquely defined by the property that
$$\int_{\...
- 2,517
2
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0
answers
130
views
How do you solve this problem of square numbers?
Whether there are different positive integers
$a_1,a_2,a_3,b_1,b_2,b_3$ , such that $a_ib_j+1\ (1\leq i,j\leq3)$ is a perfect square
number?
I think there is an infinite number that satisfy this ...
- 273
2
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answers
328
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Decomposition group $G_p$ equals the absolute Galois group of the p-adics
I am reading some lecture notes and came across the fact that the group $G_p \leq G_{\mathbb{Q}}$ (where $G_{\mathbb{Q}}$ is the absolute Galois Group of the rationals and $G_p$ is the decomposition ...
- 313
2
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56
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Sum equal to product in $\mathbb{Z}/n\mathbb{Z}$ under constraints
I need to solve the following problem.
Let $A=\{a_0,a_1,a_2,\cdots,a_k\}$
Find $a_i$ such that:
$|A|\geq 3$
$a_i\in \mathbb{N}$ and $a_i\in [32,127]$
$p=\sum_i a_i$ is prime
and
$\sum_i a_i = \prod_i ...
- 314
2
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0
answers
69
views
Guessing algebraic number by its binary digits
Positive integers $d$, $H$ are given. It's known that $r\in(0,1)$ is such that $p(r)=0$ for some nonzero $p\in\mathbb Z[x]$ with $\deg p\leq d$ and coefficients $c_i$, such that $|c_i|\le H$, $i=0,\...
- 235
2
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0
answers
39
views
Find the largest number of natural numbers to put around the circle to have sum of all numbers: 800 and sum of remainders of adjacent numbers: 500
n natural numbers were put around the circle. The sum of all numbers is equal to 800. For each pair of adjacent numbers, the larger was divided by the smaller with ...
2
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0
answers
133
views
How sparse/"thin" (asymptotically) can additive bases of order $2$ be?
A subset $B$ is called an (asymptotic) additive basis of order $2$ if every sufficiently large natural number $n$ can be written as the sum of at most $2$ elements of $B.$
How small/sparse can such ...
- 20.2k
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33
views
For half integral weight newform does the Shimura correspondance give a newform?
in number theory,
half integral wieght newforms implies the shimura correspondance is it also newforms ?
namely,
if $f(z)=\sum_{n=1}^{\infty}a(n)q^n\in S_{k+1/2}^{new}(N,\chi)$ is half integral ...
- 21
2
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0
answers
55
views
Coinvariants of rigid meromorphic function (on the p-adic upper half-plane)
Maybe someone can help me out. I am considering the p-adic upper half-plane $\mathcal{H}_p$ given on points by $\mathbb{P}^1(\mathbb{C}_p)\setminus \mathbb{P}^1(\mathbb{Q}_p)$, viewed as a rigid ...
2
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0
answers
160
views
$n\mid5^n - 3^n + 1$. Does it imply liminf of $\nu_3(n)$ to go to infinity?
Suppose $S$ is a set of all positive integers $n$ such that $n\mid5^n-3^n+1$.
For each nonnegative integer $m$ define $S_m$ as $\{n\in S:\nu_3(n)=m\}$, where $\nu_p(n)$ is $p$-adic valuation of $n$ (i....
- 235
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0
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84
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Is the Mobius Inversion Theorem an if and only if?
The Mobius inversion theorem was presented to me as follows:
Given $F: \mathbb{z}_{>0} \longrightarrow \mathbb{c}$ where $F(n) = \sum_{d\mid n} f(n)$ then $f(n) = \sum_{d\mid n} F(n) \cdot \mu(n/d)$...
- 133
2
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0
answers
101
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Contragredient of an automorphic representation
I found that I didn't quite understand how to think about the contragredient of an automorphic representation. I have read this post on Mathoverflow, which is helpful:
https://mathoverflow.net/...
- 1,407
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0
answers
78
views
Is this proof by contradiction correct?
In proving that $\mathbb{Z}/n$ is a field iff $n$ is prime, I found the following proof by contradiction of the $\Rightarrow$ implication and was wondering if it is correct.
Assume $\mathbb{Z}/n$ is ...
- 3,322
2
votes
0
answers
138
views
Could this yield a formula for the Partition numbers?
Background: Lately, I have fallen down the rabbit hole of partition numbers. Specifically the partition function, $p(n)$. It's well known that no closed-form expression (with only finitely many ...
- 4,462
2
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0
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70
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When is $\sum_{1 \leq n \leq k}n^{-n+k}$ prime?
Consider the following finite sum $f: \mathbb{N} \rightarrow \mathbb{N}$ defined as
$$f(k) = \sum_{1 \leq n \leq k}n^{-n+k}$$
$$ = 2 + 2^{k-2} + 3^{k-3}...+ (k-1)$$
It is easy to see that $f(2) = 2$ ...
user1107557
2
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answers
51
views
The power for which the product of the digits is $n$ greater times the sum of the digits
Find the smallest integer $n⩾2$ such that for some positive integer $k$ the product $p(n,k)$ of the digits of $n^k$ is equal to $ns(n,k)$ where $s(n,k)$ is the sum of the digits of $n^k$.
I considered ...
- 2,344
2
votes
1
answer
62
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shifted set of residue classes hitting intervals of length p/k Alon and Spencer problem 4.8.6
I am working on the problem 4.8.6 from Alon and Spencer and I am completely stuck. The problem is the following:
We have a set $X$ of at least $4k^2$ residue classes modulo a prime $p$. We want to ...
- 433
2
votes
0
answers
90
views
Sign of fundamental unit in real quadratic number fields with 1 mod 4 discriminant factors
Let $K$ be a real quadratic number field of discriminant $D$ with fundamental unit $\varepsilon$. Further, I want to assume that each positive factor $n$ of $D$ satisfies $n \equiv 1 \pmod 4$. (For ...
- 6,126
2
votes
0
answers
114
views
Intersecting a lattice with a linear subspace.
Let $ W $ be a linear subspace of $ \mathbb{R}^n $.
Let $ \Gamma $ be the $ \mathbb{Z} $ span of some set of linearly independent vectors $ v_1, \dots, v_k $ in $ \mathbb{R}^n $.
Is there any ...
- 8,178
2
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0
answers
48
views
Using summation of Euler Phi Function to calculate the number of possible positive slopes in a n x n grid
Background:
Suppose I have a n x n grid in the Cartesian plane with the origin at the bottom left-hand corner and I wish to find the number of co-prime integer pairs in this n x n grid. The reason for ...
- 21
2
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0
answers
68
views
Trace of norm ideal
Let $I$ be a fractional ideal of a real quadratic number field $K$ of discriminant $D$. I thought a little bit about traces of ideals and want to ask if the following is correct. I have a proof for ...
- 6,126
2
votes
0
answers
73
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Are there infinitly many prime numbers in this sequence?
Consider the sequence
$$S_n=1+n!\sum_{k=1}^{n} k$$
Some values of n such as 1,2,3 and 4 give prime outputs. Others, such as n=11 give a composite number. The question is are there infinitely many n ...
- 51
2
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0
answers
153
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Who wins in this simple "factoring game" depending on the starting number?
There is a given $N$ written on a board. Two players, Alice and Bob choose a number from the board and factorize that number to $N=XY$ where $\gcd(X,Y)\neq1$, then erase $N$ and write $X, Y$ on the ...
- 742
2
votes
0
answers
200
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Count the cusps of $\Gamma_0(N)$.
I am reading Shurman's A First Course in Modular Forms, and I got stuck in the argument to count cusps of $\Gamma_0(N)$ in page 103, here is the full argument for reference:
To count the cusps of $...
- 657
2
votes
0
answers
78
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Measuring how good the best possible rational approximation of a real number is
The Markov constant of a real number $r$ is
$$
\limsup_{d \to \infty} \frac{1}{|r-n/d|d^2}
$$
where we choose the best possible $n$ for each corresponding $d$, e.g. $n = \text{round}(r\cdot d)$.
This ...
- 7,878
2
votes
0
answers
86
views
Solutions of $(x^p - y^p)/(x - y) = z^p $
I want to show that the diophantine equation
$ (x^p - y^p)/(x - y) = z^p $
where the prime $p > 3$, can just have trivial integer solutions $\{x, y, z\}$ like $\{1, -1, 1\}$
I used the theorem $IV$ ...
- 743