# 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

10,451 questions

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33 views

### A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime.
Conjecture. Let $p$ be an odd ...

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39 views

### A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$

On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...

**4**

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**1**answer

87 views

### Near-Legendre Conjecture

Ingham has shown that there is a prime between $n^{3}$ and $(n+1)^{3}$ for large enough $n.$
Legendre's conjecture about the existence of primes between consecutive perfect squares is of course open....

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**2**answers

112 views

### Numbers of the form $2^ma + 2^nb$ where $\text{gcd}(a,b) = 1$

Given positive integers $a,b\in\mathbb{N}$ with ${\text gcd}(a,b) = 1$, and given a positive integer $d$, are there necessarily positive integers $m,n$ such that $d \;| \; (2^ma + 2^nb)$?

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72 views

### Probable primes of a particular form

Concatenating two consecutive Mersenne numbers in base 10, I found these probable primes:
$(2^{215}-1)*10^{65}+2^{214}-1$
$(2^{69660}-1)*10^{20970}+2^{69659}-1$
$(2^{92020}-1)*10^{27701}+2^{92019}-...

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37 views

### Collection of equivalent forms of a precise statement: the number of elements of a well defined set is infinite

This main aim of this forum is to collect a "big list" of equivalent forms of number theory problems equivalent to a statement of the form:
The case (A) is true if and only if the number of elements ...

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52 views

### Are there “elementary” proofs of the openness of norm subgroups and of the norm limitation theorem?

Let $K$ be a local field and $L/K$ be a finite extension. Let $L^{ab}$ be the maximal abelian subextension of $K$ in $L$. Write $N_L$ (resp. $N_{L^{ab}}$) for the image of the norm map from $L$ (resp. ...

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**1**answer

76 views

### Rate of approximation of Legendre's constant

Roughly how big is log(n)−(n/π(n))-1 is as a function on n? It asymptotically approaches zero, but given how long it took to figure out that Legendre's constant is exactly 1 it seems like it must ...

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51 views

### Krull dimension of completions in non-noetherian setting (especially completed perfections)

What are some general results describing the Krull dimension of the completion of a non-Noetherian ring with a "nice" topology?
An example of the sort of "nice" topological ring I'm looking for is a ...

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48 views

### Bounds for the number of integers not generated by some subset of the primes

For $ p $ a prime number, $x $ a positive real number greater than $ p $ and $ k $ an even positive integer, say a maximal subset $ A $ of the primes not exceeding $ x $, containing $ p $ and ...

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61 views

### Power of an integer as a sum of $\binom{n}{n-2}$ integers

Consider the following equation
$$
y^n=\sum_{k=1}^{\frac{n(n-1)}{2}} x_k,
$$
where $x,y,n,x_k\neq 0$ are integers.
Although I found a lot of material about how to express an integer as a sum of ...

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**2**answers

194 views

### upper bound of consecutive integers which are not coprime with $n!$

Is there any research on getting upper bound of the maximal possible number of consecutive positive integers which are less than $n!$ and NOT coprime with $n!$?
Easy to see that lower bound $\ge n$, ...

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239 views

### Product of sum of reciprocals of prime numbers

For any positive integers $k$ and $\ell$, does the equation
$$\left(\sum_{i=1}^k \frac{1}{p_i}\right) \left(\sum_{j=1}^\ell \frac{1}{q_j}\right) = 1$$
have solutions in distinct primes, that is, $p_1, ...

**2**

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**1**answer

159 views

### A truncated divisor sum

I am interested in an upper bound for
$$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...

**5**

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**1**answer

233 views

### How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $G$ be a connected reductive group over a number field $K$, $P$ be a parabolic subgroup of $G$ defined over $K$, $X=G/P$ be the generalized flag variety which is a smooth projective variety over $...

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70 views

### Number of $b$-separated Sidon sets with pairwise difference set intersection bounds

Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...

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136 views

### On a theorem on the face of Hardy's book [on hold]

I wonder what's the theorem of geometry on the green facepage in the following website:
https://www.google.co.jp/search?q=a+course+of+pure+mathematics+by+g.+h.+hardy+geometry&source=lnms&tbm=...

**1**

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170 views

### Function on two variables that restricts to a polynomial

Lets say that I have a function $F(x,y)$ that is defined on nonnegative integers (or at least those are the values I care about) and is symmetric, so that $F(x,y)=F(y,x)$. Moreover, I know that for ...

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74 views

### The typical amount of lattice points in a set as dimension goes off to infinity

I have encountered a geometry-of-numbers-type problem when calculating an entropy in a lattice communications scheme:
$L^{(n)}$ is a lattice uniform over all those in $\mathbb{R}^n$ with base ...

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**1**answer

73 views

### Lowering $i$th shortest vector of a lattice

LLL guarantees that we can find a basis $v_1,\dots,v_n$ of a lattice in $\mathbb R^n$ with
$$\|v_i\|\leq \gamma_{i,n} \det(\Lambda)^{1/(n-i+1)}$$ where $\gamma_i$ is a function only of $i$ and $n$.
...

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109 views

### Field of Definition of Quotient of Elliptic Curve

In Silverman's Arithmetic of Elliptic Curves, Chapter III, Proposition 4.12, we have the statement that if $E/F$ is an elliptic curve and $\Phi$ is a $\mathrm{Gal}(\bar{F}/F)$-invariant subgroup then ...

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187 views

### Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...

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176 views

+50

### Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup:
Let $m\geq 1$ be an integer, $\mathbb{F}$ be a finite field of characteristic $p$ and $W(\mathbb{F})$ the ring of Witt-vectors with residue field $\mathbb{F}$ and $\sigma:W(\mathbb{F})\...

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171 views

### Is 100 the only Leyland number that is a square?

Leyland numbers (named for Paul Leyland) are positive integers of the form $x^y + y^x$ , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980
I thought ...

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**1**answer

86 views

### Density of numbers with multiple factors near square root

Fix constants $1\leq \alpha<\beta$. What is the density of the set of positive integers $n$ with at least two factors between $\alpha\sqrt{n}$ and $\beta\sqrt{n}$?
(I am specifically interested ...

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**1**answer

209 views

### Conjecture about an Exponential Sum

Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum_{x \in ...

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106 views

### Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form
$$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$
$$\vdots$$
$$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$
where $h_1(x_1,\dots,x_{...

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224 views

+100

### Pisot conjugates

An informal version of my question is "If we have a Pisot number between 1 and 2 of a very large degree, is it true that all its other conjugates are very close to 1 in modulus?"
A more formal ...

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85 views

### Does $(p-1)^2$ divide $\det[(\frac{i^2+cij+dj^2}p)]_{0\le i,j\le p-1}$ when $(\frac dp)=-1$?

Let $p$ be an odd prime. As in my paper, for $c,d\in\mathbb Z$ let us define
$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}p\right)\right]_{0\le i,j\le p-1},$$
where $(\frac{\cdot}p)$ is the Legendre ...

**3**

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117 views

### Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...

**5**

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**3**answers

547 views

### When is $2\varphi(n) > n$ – and how to prove it?

When coloring the squares of the Ulam spiral not only by black and white (for being prime or non-prime) but by shades of grey representing the normalized totient function $\varphi(n)/n$
and ...

**5**

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**1**answer

208 views

### On the “infinitely often in” relation between subsets of $\mathbb{N}$

Let ${\mathbb N}$ denote the set of positive integers, let $A,B\subseteq \mathbb{N}$. For $n\in\mathbb{N}$ we set $n+A:=\{n+a: a\in A\}$. We say that $A$ is infinitely often in $B$ if the set $$\big\{...

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102 views

### Final step in Coppersmith?

In the final step in Coppersmith technique we have $n$ polynomials (possibly non-homogeneous) in $\mathbb Z[x_1,\dots,x_m]$ where $m\leq n$ and using elimination theory we extract the common integer ...

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79 views

### On a much weaker version of the Normal conjecture

I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-...

**2**

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**1**answer

156 views

### Integer valued polynomials over several variables

For simplicity this is about polynomials in just two variables.
Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials
$X^iY^j$ and therefore as a sum of polynomials $p_{ij}\...

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57 views

### Similar reduced integral matrices

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example)
For all $a,b,c\in\mathbb{Z}$ such that $ac-b^2=d,$ set $[a,b,c]_d=\begin{pmatrix}...

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73 views

### Height variation of abelian varieties within an isogeny class

Let $A$ be an abelian variety defined over a number field $K$ of dimension $g \geq 2$, and put $h_F(A)$ for the (stable) Faltings height of $A$. It is well-known from the seminal paper of Faltings ...

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**1**answer

152 views

### On the determinant $\det[(\frac{i^2+dj^2}p)]_{0\le i,j\le(p-1)/2}$ with $(\frac dp)=-1$

Let $p$ be an odd prime. For $d\in\mathbb Z$ we define
$$T(d,p):=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{0\le i,j\le(p-1)/2},$$
where $(\frac{\cdot}p)$ is the Legendre symbol.
By (1.17) of my ...

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**1**answer

123 views

### To what extent does the $(\mathfrak{g},K_{\infty})$ module determines the automorphic representation?

In this question, we will follow the notations and definitions from the book "Automorphic Representations and L-Functions for the General Linear Group" by Goldfeld and Hundley. For simplicity, let's ...

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410 views

### Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)

Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?

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603 views

### Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?

**3**

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170 views

### Algebraic points on a curve with small degree

Let $d \geq 2$ be a positive integer, and let $K_d$ denote the compositum of all fields of degree $d$ over $\mathbb{Q}$.
Let $Y$ be an algebraic curve defined over the rationals and has genus $g \...

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118 views

### On sets of coprime integers in intervals

Briefly,
Question: Is it "good enough" to use least prime factor in choosing a maximal set of coprime integers in an interval?
The post title comes from a 1993 paper of Erdos and Sarkozy. They ...

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111 views

### When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?

Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...

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95 views

### What does it mean for a complex valued function on $G(\mathbb A)$ to be smooth (or smooth of compact support)?

Let $G$ be a linear algebraic group over a number field $k$. Let $\mathbb A$ denote the adeles of $k$, $\mathbb A_f$ the finite adeles, and $k_{\infty} = \prod\limits_{v \mid \infty} k_v$. Here are ...

**46**

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**5**answers

4k views

### Jean Bourgain's Relatively Lesser Known Significant Contributions

A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture ...

**-2**

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**1**answer

55 views

### Approximation for $ \inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ by minimizing a distance

Under Goldbach's conjecture, let $r_{0}(n) : =\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\} $ and $k_{0}(n) : =\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $. The PNT implies that one can expect to have $ \dfrac{...

**11**

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**3**answers

342 views

### Prove that $\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$ with $p$ being an odd prime

First, I have to admit that I have already asked the same question on MSE several days ago. If I am bending any rules, I apologize for that and moderator can delete or close this question without ...

**4**

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**1**answer

243 views

### Proper Way To Compute An Upper Bound

I regard to the proof of Lemma 10 in "A remark on a conjecture of Chowla" by M. R. Murty, A. Vatwani, J. Ramanujan Math. Soc., 33, No. 2, 2018, 111-123,
the authors used the average value $(\log x)^...

**6**

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**0**answers

203 views

### Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by
$$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$
In 2004, R. Chapman [Acta ...