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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

3
votes
0answers
154 views

A conjectural lower bound for $|\{\sum_{k=1}^nka_k:\ a_1,\ldots,a_n\ \text{are distinct elements of }\ A\}|$

Motivated by Question 315568 of mine, I'm interested in the set $$S(n):=\bigg\{\sum_{k=1}^n k\pi(k):\ \pi\in S_n\bigg\}.$$ It is easy to see that $$S(1)=\{1\},\ S(2)=\{4,5\}\ \text{and}\ S(3)=\{10,...
2
votes
0answers
196 views

Algebraically independent vectors in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
1
vote
1answer
154 views

Derangements and unit fractions

Motivated by a recent question of Zhi-Wei Sun and its nice answer by Zhao Shen, here are two related questions. Let $S_n$ be the group of permutations on $\{1, 2, \ldots, n\}$. a. For each $n \ge ...
0
votes
0answers
104 views

Probability of degree $0$ gcd between every pair of random homogeneous polynomials shifted by random primes?

Take $n,d,B\in\mathbb Z_{>0}$ with $d<n$ and denote $\mathcal M_{n,d}$ to be set of all total degree $d$ monomials in $n$ variables $x_1,\dots,x_n$ with degree $\leq1$ in each variable (...
2
votes
0answers
86 views

Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
3
votes
0answers
193 views

Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?

A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that $$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ ...
8
votes
3answers
330 views

Physical Applications of Locally Symmetric Spaces

Locally Symmetric Spaces are the basis of the Langlands program—a set of ambitious and interconnected conjectures connecting representation theory to number theory, firstly proposed in 1967 by Robert ...
2
votes
0answers
68 views

Ideal class groups of real abelian number fields

I have a question from page 13 of Thaine's article on Ideal class groups of real abelian number fields - Annals of Math, 128 (1988). Here's an attempt to summarize. Let $W = E/C$ where $E$ is the ...
1
vote
0answers
79 views

Probability distribution from equidistribution - II

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
1
vote
0answers
50 views

Method of Coppersmith optimal for multivariate?

It is shown that Coppersmith method yields optimal integer root extraction for univariate polynomials in https://arxiv.org/abs/1605.08065 and a follow up work attempts this for bivariate polynomials ...
3
votes
1answer
166 views

Is there relationship between $\mu=0$ for an elliptic curve and the irreducibility of its residual representation at a prime $p$?

I remember it being mentioned at a talk that if $E$ is an elliptic curve over $\mathbb{Q}$ and $p$ a prime at which it has good reduction then the dual to the Selmer group over the cyclotomic $\mathbb{...
2
votes
1answer
127 views

How likely is it for Selmer groups to have mu invariant 0?

Given a number field $K$, how likely is it that we'll find at least one elliptic curve $E/K$ such that the $\mu$-invariant of its Selmer group is 0 (in a cyclotomic extension)?
4
votes
2answers
129 views

Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty: Specifically, ...
4
votes
0answers
154 views

Lower bound for some sums of roots of unity

Let $n$ be a positive integer (assume $n$ is prime for simplicity), and let $x_k = \pm1$, for $k = 0,1,2,..., n-1$. Let $\rho$ be an $n-$th primitive root of unity, I am interested in a lower bound ...
9
votes
1answer
338 views

Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$

In Tate's famous paper about $p$-divisible groups, for a prime number $p$ he asks whether there exists a $p$-divisible group $G$ over $\mathbb Z$ such that $G$ is not a direct sum of $\mu_{p^\infty}$ ...
8
votes
1answer
513 views

Linear equations in primes

Quoting from Green-Tao, "Linear equations in primes" (especially Cor. 1.9 in https://arxiv.org/pdf/math/0606088.pdf), any system of linear forms of finite complexity and without any local obstructions ...
6
votes
0answers
116 views

Can the partition function $p(n)$ take perfect power values?

Recall that the perfect powers are those integers $m^k$ with $k,m\in\{2,3,\ldots\}$. I don't consider $0$ or $1$ as a perfect power. Y. Bugeaud, M. Mignotte and S. Siksek [Annals of Math., 2006] ...
7
votes
1answer
171 views

Distribution of $\{cn^a\}$

Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) ...
7
votes
1answer
414 views

Is the sum $\sum_{d\mid n}\frac1{d+1}$ never integral?

Recall that a positive integer $n$ is a perfect number if and only if $$\frac{\sigma(n)}n=\sum_{d\mid n}\frac1d=2.$$ QUESTION: Is my following conjecture true? Conjecture. (i) We have $\sum_{d\mid ...
0
votes
0answers
100 views

Reference request for bounds of $n$-th composite

Motivation I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions. Recently during trying to understand and prove the ...
2
votes
0answers
53 views

An issue with showing that an Iwasawa module has zero $\mu$ invariant

Let $\chi$ denote the $p$-adic cyclotomic character. Let $\mathbb{Q}^{cyc}$ denote the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. Let $\gamma$ be the topological generator of $\Gamma=\text{...
1
vote
3answers
572 views

Question concerning twin primes

The multiplication graphs $p/q$ – with an edge between $a,b$ iff $ap \equiv b\operatorname{mod} q$ – for some twin primes $p$ and $q = p+2$ reveal an astonishing pattern: $(p+0)/(p+1)$ ...
3
votes
0answers
69 views

Independence of number fields generated by roots of Littlewood polynomials

Let $\mathcal{R}_d \subset \bar{\mathbb{Q}}$ be the set of all roots of degree $d$ polynomials with $\{-1,0,1\}$ coefficients and $$ c(d) := \min_{\substack{ \alpha, \beta \in \mathcal{R}_d \\ \alpha^...
2
votes
1answer
164 views

Does $\det[\lfloor(i^2+j^2)/p\rfloor]_{1\le i,j\le(p-1)/2}$ vanish for each prime $p>7$ with $p\equiv3\pmod4$?

Let $\lfloor x\rfloor$ be the floor function. QUESTION: Does the determinant $$D_p=\det\left[\left\lfloor\frac{i^2+j^2}p\right\rfloor\right]_{1\le i,j\le(p-1)/2}$$ vanish for each prime $p>7$ with ...
0
votes
0answers
130 views

Find all placement methods are made so that $S=\sum\limits_{1\leq i<j\leq n}|P_iP_j| ^2$ takes the maximum value

A few days ago, China held the Mathematics Olympiad. In this competition, all the students did not do the last question. But what I'm more interested in right now is, what's the background to this ...
2
votes
0answers
108 views

Chen primes and permutations

In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes. For $...
1
vote
0answers
86 views

Roots of unity, vanishing sums and derivatives

Fix integers $n\geqslant1$ and $k\geqslant 0$. For an integer $i$, the $k$-fold derivative of $x^i$ can be denoted by $i^{\underline{k}}x^{i-k}$ where $i^{\underline{k}}$ means $i(i-1)\cdots(i-k+1)$ ...
19
votes
1answer
613 views

Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$

Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$. Motivated by Question 315568 (http://mathoverflow.net/questions/315568), here I pose the following question. QUESTION: Is ...
8
votes
1answer
136 views

Index of the endomorphism ring of an abelian surface

For an abelian surface $A/\mathbb{Q}$ such that $R:=\mathrm{End}_{\mathbb{Q}}(A)$ is an order in a real quadratic field $K$ (so a $\mathrm{GL}_2$-type surface), is there a bound on the index $[O_K : R]...
2
votes
0answers
84 views

What is the current status on the corank conjecture for Selmer groups (2)?

This is a follow up to What is the current status on the corank conjecture for Selmer groups? Let E be an elliptic curve over a number field $K$ an imaginary quadratic field in which a prime $p$ ...
0
votes
0answers
63 views

Meyer's class number formulas

In my previous question I asked for a reference for $L$-series of quadratic orders in connection with a certain class number formula. It seems that this had been investigated by Curt Meyer. Is there ...
5
votes
0answers
118 views

List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$

The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
0
votes
0answers
115 views

Permutations $\pi\in S_n$ with $p_k+p_{\pi(k)}+1$ prime for all $k=1,\ldots,n$

As usual, let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$. QUESTION: Is my following conjecture true? Conjecture. For any positive integer $n$, there is a permutation $\pi\...
5
votes
1answer
269 views

What is the current status on the corank conjecture for Selmer groups?

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ a prime. It is conjectured in the book of Coates and Sujatha "Galois Cohomology of Elliptic Curves" (Conjecture 2.5) that the corank of the ...
1
vote
1answer
208 views

Odd permutations $\tau\in S_n$ with $\sum_{k=1}^nk\tau(k)$ an odd square

For any positive integer $n$, as usual we let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$. QUESTION: Is it true that for each integer $n>3$ there is an odd permutation ...
4
votes
1answer
75 views

Separate the trivial partition by a linear hyperspace

Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that: $\langle a,e\rangle=0$ and for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\...
1
vote
0answers
80 views

Dirichlet series of Euler's totient function for Gaussian integers?

Define the Euler's totient function for Gaussian integers $f:\mathbb Z[i]_{\ne 0}\mapsto \mathbb Z_{>0}$: $$f(z):=\sum_{\substack{q\in\mathbb Z[i]_{\ne 0}\\|q|\le|z|, |\gcd(q,z)|=1}}1,$$ and the ...
2
votes
2answers
695 views

If Goldbach's Conjecture is eventually true, is it necessarily true?

We have all heard that if Goldbach's conjecture is independent, then it is true. This is because if GC is false then there is an even number which is not the sum of two primes, and hence a finite ...
3
votes
0answers
90 views

Integer partitions under divisibility constraint

Consider integer partitions of $x \in \mathbb{N}$ of size $k$ under the constraint that the partition elements are distinct and the ratio of any element to each smaller element is a natural number. ...
6
votes
1answer
191 views

Integral Tate-Sen theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
31
votes
1answer
1k views

Probably true, but provably unprovable

I'm wondering if anyone has found, or can find, a sequence of statements $P(n)$ ($n \in \mathbb{N}$) such that: Heuristic arguments using probability theory suggest that all the statements $P(n)$ are ...
1
vote
0answers
99 views

Is there a (Riemann) explicit formula for $\sum_{p\le x}\frac{1}{p}$ involving a sum over the non-trivial zeros ρ of the Riemann zeta function?

Let $f(x)=\sum_{p\le x}\frac{1}{p}$ and $f_0(x)=\frac{1}{2}(f(x+0)+f(x-0))$. Then is there a (Riemann) explicit formula for $f_0(x)$ involving a sum over the non-trivial zeros ρ of the Riemann zeta ...
6
votes
1answer
119 views

Explaining patterns in modular multiplication graphs

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look ...
4
votes
0answers
79 views

How to determine the unramified character corresponding to an unramified Langlands parameter?

Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
11
votes
0answers
218 views

Easy cases of Herbrand's theorem

$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$ I recall Herbrand's theorem about class groups of cyclotomic fields: Let $p$ be an odd prime, let $\zeta$ be a primitive $p$-th root of $1$ and let $K = \QQ(\...
15
votes
5answers
1k views

Sum of the reciprocals of radicals

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$. For a paper, I need the result that $$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\...
5
votes
0answers
115 views

A relation concerning the “sum of squares” counting function $r_2(n)$

This is a re-post from MSE as I did not get any response there. Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...
17
votes
1answer
2k views

A mysterious connection between primes and squares

Motivated by two previous questions of mine (cf. Primes arising from permutations and Primes arising from permutations (II)), here I ask a curious question which connects primes with squares. ...
7
votes
0answers
158 views

Number fields ordered by discriminant

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
3
votes
0answers
262 views

Primes arising from permutations (II)

In Question 315259 (cf. Primes arising from permutations) I asked a question on primes arising from permutations which looks quite challenging. Here I pose a new question in this direction which does ...