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
15,946
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4
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Asymptotic for number of partitions of $n$ into $k$ squares, uniform in $n,k \to +\infty$
Let $p^{(s)}(n)$ be the number of ways of writing the positive integer $n$ as a sum of perfect $s$-powers, where the order does not matter. For example, $p^{(2)}(9) = 4$ since
$$9 = 1^2 + 1^2 + 1^2 + ...
CommunityBot
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0
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364
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Recognizing the Galois group from the field discriminant
Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
- 2,643
9
votes
1
answer
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Are there infinitely many primes of this form?
The semiprime $87 = 3*29$ has a curious property: it's the fact that both
$87^2 + 29^2 + 3^2 = 8419$
and
$87^2 - 29^2 - 3^2 = 6719$
are prime numbers.
This intrigued me and led me to wonder if ...
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29
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4
answers
3k
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Is there an 11-term arithmetic progression of primes beginning with 11?
i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
- 4,686
5
votes
1
answer
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Does every prime $p$ appear in a $p$-term arithmetic progression of primes? [duplicate]
This is a follow-up to an earlier question.
The answer to that question was found on this page. The discussion on OEIS seems to suggest that, for any prime $p$, there should exist a $p$-length ...
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-2
votes
1
answer
575
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Can this criterion to indicate the randomness some numbers? [closed]
John Derbyshire in his book PRIME OBSESSION says on page 366:
CHAPTER 3
10.
"Here is an example of e turning up unexpectedly. Select a random number
between 0 and 1. Now select another and add ...
- 1,305
104
votes
6
answers
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Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
CommunityBot
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0
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88
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Infinite difference length of integer subsets
Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...
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2
votes
0
answers
255
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The Hilbert Symbol and real algebraic geometry
Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have:
$$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...
- 99.1k
0
votes
1
answer
122
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On the asymptotics of a certain sum involving the prime counting function
Let $\pi(x)$ denote the number of primes $\leq x$. What is the asymptotic form for
$$\sum_{r=1}^{\pi(x)-1} \Bigg(\frac{(\pi(x))!}{r!(\pi(x)-r)!}-\frac{(\pi(x))!}{(r-1)!(\pi(x)-r+1)!} \Bigg) $$ ?
The ...
- 12.7k
9
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1
answer
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Are all totally ramified $\mathbb{Z}_p$-extensions of local fields come from (relative) Lubin-Tate formal groups?
The setup is as follows:
$k/\mathbb{Q}_p$ is a finite extension, $\mathfrak{p}$ is the maximal ideal of $\mathcal{O}_k$, $q=\#(\mathcal{O}_k/\mathfrak{p})$
$k'/k$ is a finite unramified extension of ...
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1
answer
849
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A curious valuation of this sequence
The sequence $a_n$ given by
$$a_n=\sum_{k=0}^n\frac{n!}{k!}$$
is found at A000522 on OEIS with a description: total number of arrangements of a set with $n$ elements. Let $\nu_2(x)$ denote the $2$-...
- 33.2k
4
votes
1
answer
436
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Density of integers with a large rough divisor
Let $N < 2^a$ be a positive integer chosen uniformly at random. Let $\tilde{N}$ be the result of removing from $N$ all its prime factors less than $2^b$. What is the probability that $\tilde{N}$ is ...
14
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1
answer
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Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$
Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$
So we must have
$$2^{\frac{p-1}{4}}\equiv \...
- 20.5k
1
vote
1
answer
256
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Sum of log over friables
Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$\sum_{\substack{n \leq x \\ P(n)...
- 1,193
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votes
4
answers
2k
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How do these primes jump?
Update 2017.08.28: I am still looking for references. I have posted a request to https://cs.stackexchange.com/q/79971 which includes some literature references I found which are of interest but still ...
- 13k
5
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0
answers
767
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The Grimm Machine(s): A Collatz Conjecture Rival?
Edit 2018.08.08 This answer https://mathoverflow.net/a/307881 will be updated to give recent information about S, especially a forthcoming preprint. End Edit 2018.08.08
Just as the Collatz ...
- 13k
3
votes
0
answers
171
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Extension of Galois representations from geometry
Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of ...
- 1,378
2
votes
0
answers
126
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Expressing modular functions of level 9 and 32 as rational functions
Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
- 2,335
1
vote
1
answer
138
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Cryptography with general RSA type integers?
Denote $\mathcal N_r=\{n\in\mathbb Z:\exists\mbox{ distinct equal bit primes }p_1,\dots,p_r:n=p_1p_2\dots p_{r-1}p_r\}$.
$\mathcal N_1$ refers to primes and $\mathcal N_2$ referes to balanced ...
- 10.1k
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1
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Books with exercises to learn Langlands program, Galois representations, modular forms
I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
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1
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Computational complexity of finding the class number
Let $f(x)$ be an integral irreducible monic polynomial and $\alpha$ be its root. What is the computational complexity of finding representatives of ideal classes of the integral ring $\mathbb{Q}[\...
- 5,709
6
votes
1
answer
563
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An asymptotic formula for this sum
Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.
$$\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$
where $a$ and $b$ are two coprime ...
- 178k
8
votes
1
answer
391
views
Goldbach's conjecture for the Liouville function
Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.
2
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1
answer
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Centralizer of Shimura datum defining a Shimura curve in $A_2$
Let $B$ be an indefinite quaternion algebra over the rationals, let $G$ be the reductive algebraic group defined by $G(A) = (B\otimes A)^*$ for ${\bf Q}$-algebras $A$; hence $G({\bf R}) = GL_2({\bf R})...
- 6,141
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12
answers
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Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 6,197
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votes
8
answers
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The inverse Galois problem, what is it good for?
Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience ...
- 25.5k
2
votes
1
answer
395
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Are all unitary perfect numbers divisible by 6?
I recently learned that numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of the unitary divisors of $n$, are called unitary perfect numbers, where a divisor $d$ is a unitary ...
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votes
1
answer
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Fluctuations of Liouville function
One of my friend (who is working in mathematics) was asking the following question. Let us take Liouville λ(n) function.
et S={ λ(1), λ(2), λ(3), ..... } . Then every finite length (say l) ...
- 99.1k
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0
answers
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$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$?
Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt-p)$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I came ...
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1
answer
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Extension by harmonics
Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \mathbb{C}$, does there exist ...
- 53.6k
3
votes
1
answer
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Number of $k$-free integers of bounded radical
Let $n \in \mathbb{N}$. Define the radical $R(n)$ of $n$ by
$$\displaystyle R(n) = \prod_{p | n} p.$$
In other words, $R(n)$ is the largest square-free number which divides $n$.
For an integer $k \...
- 43.3k
13
votes
1
answer
595
views
A recursive formula
$$a_0 = 1, \ \ a_{n+1} = \ 1+\frac{n *a_n}{n+a_n} , \ \ n=0,1,2,3,4,...$$
I have built the above recursive formula. Some terms of sequence are:
1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/...
- 3,696
2
votes
0
answers
58
views
Configurations of minimal vectors for a 4-dimensional symplectic lattice
The possible configurations of minimal vectors for a 4-dimensional lattice are known for ages, but what about symplectic lattices ? If a 4-dimensional symplectic lattice $\Lambda$ has two minimal ...
-4
votes
1
answer
169
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Covering system of congruences with specific properties?
A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$, ($r\geq2$) is called a covering system of congruences if every integer belongs to at least one of the residue classes, ...
- 841
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1
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Understanding a valuation property of function fields
I came across this point in a paper recently and I'm having difficulty seeing why it's true. Any explanations or hints would be appreciated.
For any prime $\mathfrak{p}$ of $\mathbb{F}_q [t]$ such ...
- 20.8k
6
votes
1
answer
891
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Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension
Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic ...
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2
answers
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Guidelines to prove that $2^{\sqrt{2}}$ is a transcendental number? [closed]
After a small search that I did I was unable to spot any answers here.What I am trying is to prove why the $2^{\sqrt2}$ is transcendental number. I know that this probably is a closed problem and ...
- 4,618
1
vote
1
answer
217
views
constructing a covering system of congruences?
A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, ...
- 4,618
6
votes
1
answer
298
views
Can a primitive root-permutation of $A=\{1, 2, \ldots, p-1 \}$ be a cycle of length $p-1$ only for finitely many $p$?
Let $p$ be an odd prime, $g$ a primitive root of $p$ and $A=\{1, 2, \ldots, p-1 \}$.
Obviously, $\sigma_g(p)=\begin{pmatrix}
1 & 2 & \ldots & {p-1} \\
g^1\pmod{p} & g^2\pmod{p} &...
- 14.3k
4
votes
0
answers
289
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power series and roots of unity
Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
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1
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Unique factorisation of prime geodesics?
In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
- 95.6k
3
votes
2
answers
617
views
Number of ways to write an integer as a sum of squares modulo $k$
Given a natural number $n$ and an element $k \in \mathbb{Z}_n$, how many solutions are there in $\mathbb{Z}_n$ to the equation $x^2+y^2 =k$? That is, I'm wondering whether there is a mod-$n$ version ...
8
votes
2
answers
417
views
Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?
Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O_F$. Let's put $B=\operatorname{Spec } O_F$, then an Arakelov divisor is an element of:
$$Div(X)\times \bigoplus_\...
- 12.8k
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votes
2
answers
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Binary representation of powers of 3
I asked this question at Mathematics Stack Exchange but since I didn't got a satisfactory answer I decided to ask it here as well.
We write a power of 3 in bits in binary representation as follows.
...
- 99.1k
9
votes
0
answers
210
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Surjectivity of reduction for Hilbert modular forms
Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$.
Then one can form the space $S_k(\mathfrak{n},\...
- 562
4
votes
1
answer
165
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The number of solutions of the equation $ax_1x_2+by_1y_2=n$
The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive ...
- 99.1k
4
votes
1
answer
527
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Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$
Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$
It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality?
EDT 1: A possible answer is Analysis of the ...
- 11.9k
10
votes
1
answer
707
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Why are the coefficients of the modular equation so large?
The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we ...
- 178k
6
votes
1
answer
594
views
Upperbounding a sum of Legendre-Symbols
Let $p$ be a prime with $p\equiv 3 \mod 4$, for any $\mathcal{I} \subset \lbrace 0,...,p-1 \rbrace $ and any $\mathcal{J} \subset \lbrace 0,...,p-1 \rbrace $ with $\vert\mathcal{I}\vert \leq \sqrt{p} $...
- 43.3k