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
15,637
questions
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Is the integer factorization into prime numbers normally distributed?
Let $P_1(n) := 1$ if $n=1$ and $\max_{q|n, \text{ }q\text{ prime}} q$ otherwise, denote the largest prime divisor of $n$.
Let us define some rooted trees $T_{n,m}$ for $1 \le m \le n$ by:
$T_{n,m}$ ...
- 4,226
13
votes
1
answer
364
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Four new series for $\pi$ and related identities involving harmonic numbers
Recently, I discovered the following four new (conjectural) series for $\pi$:
\begin{align}\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^k\binom{3k}k}{k(3k-1)(3k-2)\binom{2k}k\binom{4k}{2k}}&=\frac{3\pi}2,\...
- 173k
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115
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Does there exist an $L$-function for any subset of $\mathbb{N}$?
Consider the following prime sum:
\begin{aligned}
\sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}}
\end{aligned}
whose spikes appear at the Riemann $\zeta$ zeros as shown here.
Taking these detected spikes (...
- 1,873
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0
answers
173
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On fifth powers forming a Sidon set
We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct.
Erdős conjectured ...
- 661
6
votes
3
answers
661
views
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 ...
- 121
2
votes
0
answers
100
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Computing the Dieudonné module of $\mu_p$ from Fontaine's Witt Covector
In Groupes $p$-divisibles sur les corps locaux, Fontaine introduced a uniform construction of Dieudonné modules through the definition of the Witt covector. Consider a perfect field $k$ of ...
- 10.6k
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1
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240
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Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...
- 7,961
3
votes
0
answers
46
views
$R$-recursion for the A249833 (similar to A235129)
Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx
$$
The sequence begins with
$$
1, 1, 2, 7, ...
- 3,018
3
votes
1
answer
646
views
Why is the congruent number problem open?
I was reading up about the congruent number problem.
One of the theorems on the subject says how the two things are equivalent: a positive integer $n$ being a congruent number and elliptic curve $y^...
0
votes
1
answer
179
views
Is there any use of logarithmic derivatives of modular forms?
Does taking the logarithmic derivative of a modular form have any uses, such as identifying patterns in its coefficients or possible zeros of its corresponding L function?
- 49.1k
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0
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98
views
What is the logic behind the Extended Euclidean Algorithm procedure? [closed]
Thank you beforehand for reading my question.
In the terms that I'd want to understand the Extended version of the Euclidean Algorithm, I understand the Euclidean Algorithm as follows:
You find the ...
- 4,588
4
votes
0
answers
240
views
An algebraic version of the implicit function theorem for integers
$ \def \x {\boldsymbol x}
\def \a {\boldsymbol a}
\def \Z {\mathbb Z} $
The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
- 59.4k
4
votes
0
answers
108
views
Taking integer values of a sequence of Beurling primes
Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
- 41
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131
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what is the current best estimation for the upper bound of the exponential sum for an arbitrary irrational number $\alpha$
I would like to know what the current best estimation for the upper bound of the exponential sum
$$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\...
- 543
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votes
0
answers
381
views
+50
Genus of a number field
I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as
$$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
4
votes
0
answers
76
views
Buchi's conditional proof of the non-existence of finite algorithm to decide solubility of system of diagonal quadratic form equations in integers
I am doing some literature review regarding Buchi's problem. In particular, I am reading the relevant section in this survey paper by Mazur (Questions of Decidability and Undecidability in Number ...
- 21.7k
5
votes
0
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469
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Theorem 7.11 in Scholze's $p$-adic Hodge Theory
I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below:
Let $...
- 51
2
votes
1
answer
187
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'$\times$' or '$\otimes$' when writing $L$-functions?
Recently, I came across the Langlands correspondence theorem, there is the following line:
$$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$
where $\sigma$ and $\tau$ are ...
- 95.7k
2
votes
0
answers
84
views
$R$-recursion for the A235129
Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A'(x) = 1 + A(x)\exp(A(x))
$$
The sequence begins with
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 3,018
7
votes
1
answer
353
views
Large sets of nearly orthogonal integer vectors
This question is motivated by the Question 5 from the 2017 Asia Pacific Mathematical Olympiad. To paraphrase, the question asks what is the largest cardinality of a set $S \subset \mathbb{Z}^n$ such ...
CommunityBot
- 1
2
votes
1
answer
238
views
Which algebraic groups are generated by (lifts of) reflections?
$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem
states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, ...
- 59.4k
-6
votes
0
answers
125
views
How is research on Modular Forms usually conducted? [closed]
I have just started working on modular forms through an introductory course and let led me to think how is research on them conducted. For example, what exactly do you research about modular forms, is ...
3
votes
1
answer
393
views
Pushforward of functions on a frame bundle
Apologies in advance for the long setup and question.
Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have
$$ p_*\mathcal{O}_{Fr(L)} =...
CommunityBot
- 1
4
votes
0
answers
121
views
Sequence of digits of powers of two
Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
- 21.7k
8
votes
1
answer
661
views
The tightest prime zipper
Define a prime zipper as an increasing function $f(n)$ mapping $\mathbb{N}$ into $\mathbb{N}$
with the property that, for every $n \ge 1$, there is at least one prime within the
inclusive interval $[ ...
- 1,324
6
votes
2
answers
711
views
Raising positive integer to $c\in\mathbb{R}-\mathbb{N}$ rarely gives an integer!
Problem: Let $c>0$ be a real number, and suppose that for every positive integer $n$, at least one percent of the numbers $1^c,2^c,3^c,\dotsc,n^c$ are integers. Prove that $c$ is an integer.
My ...
- 10.6k
2
votes
2
answers
562
views
Is it true that there always exists a positive integer $n$ such that $p \mid \lfloor k^n\cdot\alpha\rfloor$?
Let $k,M$ be positive integers such that $k−1$ is not squarefree. Prove that there exist a positive real number $\alpha$, such that $\lfloor\alpha\cdot k^n\rfloor$ and M are coprime for any positive ...
CommunityBot
- 1
5
votes
2
answers
391
views
On the number of distinct prime factors of $p^2+p+1$
Is it true that, for each positive integer $c$, there exists a prime number $p$ such that $p^2+p+1$ is divisible by at least $c$ distinct primes?
CommunityBot
- 1
2
votes
2
answers
242
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 167
2
votes
0
answers
159
views
Can all modular forms be written as Eta Quotients?
I have been going through a couple of introductory courses in modular forms and am quite curious whether all modular forms can be written as eta quotients of the Dedekind eta function?
7
votes
1
answer
356
views
Bounding the growth of rational bivariate polynomials from below
The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be ...
- 4,588
5
votes
0
answers
209
views
Video abstracts for mathematical papers
I recently published a video abstract of a manuscript of mine (number theory), finding that more people are interested in its content than when I uploaded the preprint on arXiv.
Now, my main question ...
- 883
4
votes
2
answers
537
views
Computing hypergeometric function at 1
I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
- 1,514
15
votes
2
answers
2k
views
sum of three cubes and parametric solutions
The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
3
votes
0
answers
81
views
Is there a closed form for the Rudin-Shapiro sequence?
The Rudin-Shapiro sequence is defined as follows:
Let $a_n=\sum\epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2...$ are the digits in the binary expansion of $n$. $WS(n)$, the $n$th term of the ...
- 131
5
votes
0
answers
359
views
On the shortest open cubic equation
The question is: are there any integers $x,y,z$ such that
$$
1+4 x^3+x y^2+2 y z^2 = 0 \quad\quad\quad\quad (1)
$$
The motivation is: Define the length of a polynomial $P$ consisting of $k$ monomials ...
- 4,905
65
votes
1
answer
6k
views
Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
- 4,905
3
votes
0
answers
82
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 553
8
votes
1
answer
580
views
Arithmetic sequences and Artin's conjecture
(Sorry if this is a naive question; it is not my area!)
Consider the following strengthening of Artin's conjecture on primitive roots (and Dirichlet's theorem) for the case of $n=2$: every arithmetic ...
- 76.8k
53
votes
1
answer
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
- 2,468
8
votes
1
answer
586
views
Does this partial sum over primes spike at all zeta zeros?
Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \
p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines:
Below ...
- 1,001
0
votes
0
answers
57
views
Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?
What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$,
every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
- 19.1k
-1
votes
0
answers
359
views
Are these finite semirings known?
I am trying to prove the properties below, and by doing this, I hope to find a way to speed up the computation of the below defined addition and multiplication. I am also interested if these finite ...
- 2,126
2
votes
0
answers
109
views
Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 17.3k
6
votes
2
answers
601
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 95.7k
9
votes
1
answer
361
views
How fast can elliptic curve rank grow in towers of number fields?
Fix $E/K$ an elliptic curve over a number field $K$. For various towers of finite field extensions $K=K_0 \subset K_1 \subset K_2\subset\cdots$ how fast can $\operatorname{rank}(E(K_n))$ grow in ...
- 2,645
7
votes
1
answer
439
views
Sum of three squares as class numbers and Waldspurger's formula
It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
CommunityBot
- 1
-4
votes
0
answers
189
views
EC primes the heuristic says something different from the reality [closed]
Ec-primes are primes of the form $(2^n-1)\cdot 10^d+2^{n-1}-1=ec(n)$, where d Is the Number of decimal digits of $2^{n-1}$.
Up to $n=10^5$, there are 30 primes of this type.
A rough heuristic says 18 ...
- 1
16
votes
2
answers
929
views
The Stable Set Conjecture
A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation
$$n\in \mathcal S \iff dn\in \mathcal S$$
holds for almost all positive integers $n$. ...
- 1,183
3
votes
0
answers
85
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 ...
- 376