Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

8
votes
1answer
283 views

Products of Catalan numbers

Let $c(n)=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan number. It seems that a product $\prod_{n\in I} c(n)$, where $I\subset\mathbb N_{>1}$, is never a Catalan number. Is this a (known) fact?
2
votes
0answers
82 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
1
vote
1answer
220 views

Find the positive integers $x^3+y^3=3z^3$ [closed]

By Fermat Last theorem, I don't know if that's been discussed. Find all positive integers $x,y,z$ such $$x^3+y^3=3z^3$$
3
votes
0answers
129 views

I want a elaboration of the sketch of proof given in the Serre's Galois Cohomology on the existence of the dualizing module

I've wanted to understand the concept of the Dualizing module in the theory of Galois Cohomology. There are many references on it and of them all Neukirch's Cohomology of Number Fields seems to be ...
3
votes
0answers
232 views

A new combinatorial problem for finite groups

In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
4
votes
0answers
99 views

Abelian variety over Q with many roots of unity

Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...
18
votes
1answer
319 views

Curves over number fields with everywhere good reduction

My question is the following:$\newcommand{\Q}{\Bbb Q} \newcommand{\Z}{\Bbb Z}$ What is known about number fields $K$ fulfilling the condition $C_{g,K}$ "there is a smooth projective curve of ...
6
votes
1answer
609 views

Is there a 2-power-twinless prime?

Call two primes 2-power-twins if their difference is (can you guess?) a power of 2. For example, 11 and 19 are 2-power-twins. Is there a 2-power-twinless prime? I would imagine that this is doable ...
9
votes
0answers
182 views

On the permanent $\text{per}[i^{j-1}]_{1\le i,j\le p-1}$ modulo $p^2$

Let $p$ be an odd prime. It is well-known that $$\det[i^{j-1}]_{1\le i,j\le p-1}=\prod_{1\le i<j\le p-1}(j-i)\not\equiv0\pmod p.$$ I'm curious about the behavior of the permanent $\text{per}[i^{j-...
3
votes
2answers
281 views

On the sum $\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$

Motivated by Question 316142 of mine, I consider the new sum $$S(n):=\sum_{\pi\in S_{n}}e^{2\pi i\sum_{k=1}^{n}k\pi(k)/n}$$ for any positive integer $n$, where $S_n$ is the symmetric group of all the ...
2
votes
0answers
131 views

trivial solutions for Diophantine equations

Let $K$ be an odd degree number field. Consider the Diophantine equation: $$ X^4 + bY^4 =Z^2 $$ where $b\neq 0$. Say we know that the above equation has only trivial roots in $K$ (for some fixed ...
3
votes
1answer
86 views

Is coprimality in $NC$?

Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
5
votes
0answers
80 views

Relation between Faltings height and height on moduli space

Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms ...
31
votes
1answer
3k views

Does the algorithm of the Greeks produce all prime numbers?

Let ${\cal P}$ be the set of prime numbers. Define a subset ${\cal P}'=\{p_1,p_2,p_3,\cdots\}$ of ${\cal P}$ by setting $p_1=2$ and defining $p_{n+1}$ to be the smallest element of ${\cal P}$ ...
6
votes
1answer
267 views

Reference Request for a result on divisors of $p-1$

I have seen this result in several places without an English reference: There exist infinitely many primes $p$ such that $p-1=2q_1q_2$ where $q_1$ and $q_2$ are prime numbers with $q_1,q_2>p^{1/4}$...
2
votes
0answers
174 views

A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
6
votes
0answers
179 views

A challenging problem on disjoint cosets

Motivated by my negative solutions (see this paper published in Chinese Ann. Math. 13A(1992)) to two open problems on disjoint residue calsses posed by A. P. Huhn and L. Megyesi [Discrete Math. 41(...
15
votes
0answers
594 views

What is the smallest unsolved diophantine equation?

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
-2
votes
1answer
230 views

A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that $$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$. where $\zeta$ ...
0
votes
2answers
198 views

Permutations of squares and finite fields

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$, and let $$S(n)=\bigg\{\sum_{k=1}^nk^2\pi(k)^2:\ \pi\in S_n\}.$$ Motivated by Question 316142 of mine, here I ask the following ...
2
votes
1answer
91 views

Diophantine equation for generating computably enumerable set

By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is: ...
3
votes
0answers
74 views

What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post: Is this generalization of the Hopf map for quadratic field extensions surjective? I still would like to know more and, while that post got several votes,...
2
votes
3answers
209 views

Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

Denote by $\zeta$ the Riemann zeta function. It is known that $$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$ But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
6
votes
0answers
208 views

Best software to do big number calculations quickly [closed]

I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (100+ digits ). I am currently using python to test these numbers. In the ...
2
votes
0answers
89 views

What is the image of $-1$ by the local reciprocity map?

Consider the Weil group $W$ of $\mathbb{Q}_p$, that is, the subgroup of those elements of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ mapping to an integer power of Frobenius. Class field ...
5
votes
0answers
92 views

Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)

SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...
4
votes
1answer
332 views

Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem. Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...
7
votes
1answer
143 views

Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices $$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$ Let $$\pmatrix{\alpha_n & \beta_n \\...
1
vote
0answers
104 views

$\mu=0$ for elliptic curves over number fields

Greenberg conjectured that given $E/K$, there always exists $E^\prime/K$ such that $E'$ is isogenous to $E$ and $\mu(E^\prime)=0$. Michael Drinen has shown that for an elliptic curve $E/K$, it is ...
2
votes
0answers
499 views

Can the ABC conjecture be expanded?

Has anyone considered expanding the range of terms $a$ and $b$ for each $c$? I have generated triples $(a, b, c)$ that form integer triangles including the degenerate case of $a + b = c$ such that $a ...
4
votes
0answers
42 views

On the orthogonal group of a lattice on a quadratic space over dyadic local field

Let $F$ be a local field with valuation ring $R$. $V$ is a n dimensional non-singular quadratic space over $F$ with bilinear form $B$ and quadratic map $Q$. As usual, $O(V)$ denotes the orthogonal ...
21
votes
0answers
292 views

Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?

It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since $$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,...
0
votes
0answers
128 views

On segments of the series $\sum_p\frac1{p-1}$

Here I ask a question concerning segments of the divergent series $$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$ where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime. ...
1
vote
0answers
97 views

Point Counts on $G$-torsors over Finite Fields

Let's assume we have a $G$-torsor $X_{1} \to X_{2}$, where $G$ is a finite abelian group, and both $X_{1}$ and $X_{2}$ are defined over $\text{Spec}(\mathbb{Z})$. Is there an easy way to compute $\#...
0
votes
0answers
105 views

Does Coppersmith technique suffice to factor?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time? Technically I am ...
6
votes
1answer
456 views

Are the ideles literally a picard group?

I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field. Question: Is this ...
-1
votes
1answer
89 views

Discrepancy in non-homogeneous arithmetic progressions

I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies $ \left | \...
10
votes
0answers
283 views

Fourier transforms and nontrivial vector bundles

We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
1
vote
1answer
230 views

How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?
1
vote
0answers
61 views

Roots of unity and coordinates of points in abelian varieties

We consider an abelian variety $A$ defined over the rational numbers $\mathbb{Q}$. For a torsion point $P\in A(\bar{\mathbb{Q}})$, consider the field $\mathbb{Q}(P)$ obtained by adjoining to $\mathbb{...
0
votes
1answer
151 views

Is it true that $\sum_{k=m}^n\frac{\sigma(k)}k\not\in\mathbb Z$ for all derangements $\sigma\in S_n$ and $1\le m\le n$?

Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$. Recall that a permutation $\sigma\in S_n$ is called a derangemnt if $\sigma(k)\not=k$ for all $k=1,\ldots,n$. Motivated ...
5
votes
0answers
168 views

Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares. To avoid such congruence ...
0
votes
1answer
119 views

Numbers formed by the concatenation of two consecutive Mersenne numbers [closed]

The ec numbers are so defined: $ec(n)=(2^{n}-1)*10^d+2^{n-1}-1$, where $d$ is the number of decimal digits of $2^{n-1}-1$. In other words, ec(n) is obtained by the concatenation of two consecutive ...
4
votes
0answers
106 views

Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
0
votes
0answers
69 views

Relation between primeness, coprimeness, totient, and gcd function

There are two number theoretic facts that seem to be unrelated at very first sight but at second sight seem to be strongly related to each other: (1) Primeness of one number and coprimeness of two ...
4
votes
0answers
126 views

Is there a permutation $\pi\in S_n$ with $\sum\limits_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0$ for each $n>7$?

Let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$. QUESTION: Is it true that for each $n=8,9,\ldots$ we have $$\sum_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0\tag{$*$}$$ for ...
-1
votes
1answer
263 views

Odd & even permutations and unit fractions

One more motivated by recent questions of Zhi-Wei Sun. Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$. Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
-1
votes
0answers
123 views

On a certain representation for the Riemann zeta function in Montgomery-Vaughan

On page 338 of Montgomery-Vaughan's ''Multiplicative Number Theory'', there is a somewhat nice representation for the Riemann zeta function. That is, let $0<\delta\leq 1/2$. Then one has $$\zeta(1/...
2
votes
0answers
129 views

Number of primes $p$ with the same multiplicative order of 2 modulo $p$, ${\rm ord}_p\ 2$

Let $q$ be a prime number, and let $k={\rm ord}_q\ 2$, the multiplicative order of 2 modulo $q$. Is there a known upper bound (a function depending on $p$, maybe) to the number of primes $p$ such ...
2
votes
0answers
140 views

On the set $\{\sum_{k=1}^n \lambda_ka_k:\ a_1,\ldots,a_k\ \text{are distinct elements of}\ A\}$

For a field $F$ let $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\infty$ if $F$ is of characteristic zero. In 2007 I considered the linear extension of the Erdos-Heilbronn ...