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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

2
votes
0answers
32 views

Index of the congruence subgroups of $PGL_2(\mathbb{Z}_p)$

Let $\Gamma_n$ be the $n$-th congruence subgroup of $GL(2,\mathbb{Z}_p)$. So $\Gamma_n$ consists of matrices in $GL(2,\mathbb{Z}_p)$ which are congruent to the identity matrix modulo $p^n$. Let $Z(\...
5
votes
0answers
115 views

Calculating some Galois cohomology

Let $L/\mathbb{Q}$ be a Galois extension of degree $p$ and $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $p$ be a fixed prime (of good ordinary reduction if required). We use $L_\infty, \...
4
votes
1answer
204 views

How many roots of polynomial in $\mathbb Z[x]$ and $\mathbb Q[x]$ are integers on average?

Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random ...
-1
votes
0answers
36 views

The sum of logarithm rationally dependent numbers [on hold]

Let $x,y$ be logarithm rationally dependent, that is to say, $ \frac{\log x}{\log y}\in \mathbb{Q}. $ My question is that: assuming that $x\not=y$, can the sum of such number be $\frac{1}{n}$ for ...
2
votes
1answer
108 views

Questions about a certain set of primes

Let $A$ be a set of prime numbers, generated by the following procedure. Let $A_0 = \{2\}$. Let $A_n$ be generated by setting $x_n = (\prod_{p_i \in A_{n-1}}p_i) + 1$, and adding all the prime factors ...
-2
votes
0answers
98 views

Is this Diophantine equation easy to solve? [on hold]

$$(\prod_1^n 3)(2^{k-n+1}a+2^{k-n})-1=\frac{2^{k+2}-5}3b$$ $a,b,n,k\in\mathbb N$ for any a,b,n,k>0
2
votes
1answer
131 views

Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way [on hold]

I need to emulate this sequence for a program: http://oeis.org/A025302 Stuff that I've taken into account: After finding the prime divisors of a number. I take any divisor as p and apply the ...
-6
votes
0answers
73 views

Is the Polya-Vinogradov inequality on primitive character sums applicable to the Mertes function? [on hold]

The Mertens function $M(x)$ is defined as $\Big|\sum_{n\leq x} \mu(n)\Big|$, where $\mu$ denotes the Mobius function. If $\chi$ is a primitive character modulo $q$, the Polya-Vinogradov inequality ...
1
vote
2answers
168 views

On the spacing of the zeros of the Riemann zeta function

Suppose the Riemann zeta function has infinitely many zeros $\rho$ with $\Re(\rho)=\sigma$. Does it follow that for every large $T>0$, there exists some $t$ such that $T<t<3T$, where $t=\Im(\...
5
votes
1answer
251 views

What is the formality behind passing from Number Fields to Number Rings

In algebraic number theory, one constructs for each number field $K$ a subring $\mathcal{O}_K$, the integral closure of $\mathbb{Z}$ inside $K$, which carries many of the properties which make $\...
5
votes
1answer
137 views

Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$

Let $G$ be a compact abelian group. The unitary characters of $G$ form an orthonormal basis of $L^2(G)$, so every square integrable function $f: G \rightarrow \mathbb C$ admits a Fourier expansion $$...
0
votes
0answers
60 views

Independence of integer sequences arising from Dirichlet's pigeonhole

Dirichlet's pigeonhole says given sequence $a_1,\dots,a_n\in\mathbb Z$ and a prime $p$ there is an integer $t$ coprime to $p$ with $\|r_i\|\in[-p^{(n-1)/n},p^{(n-1)/n}]$ at every $i\in\{1,\dots,n\}$ ...
0
votes
0answers
34 views

Two sequence discrepancy and smallest boxes?

Take $p$ to be a prime and let $a_1,\dots,a_n\in\mathbb Z$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{ma_1}p,\dots,\frac{ma_n}p\}$$ with $m\in\{1,\dots,p-1\}...
-1
votes
1answer
109 views

If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer. In a previous post I asked if $p_n(a,b)$ was a ...
-4
votes
0answers
70 views

False constantes in first Hardy-Littlewood conjecture? [on hold]

Let $q$ be a prime number, and $m$ an even number. Let $\displaystyle\mathcal{B}_q=\{b \in \mathbb{N}^{*} \, | \, b \wedge {\small \left( \prod_{\substack{a \leq q \\ \text{a prime}}} {\normalsize a} ...
1
vote
0answers
157 views

On the determinant $\det[\sec2\pi\frac{jk}p]_{0\le j,k\le(p-1)/2}$

On the basis of my computation, I have the following conjecture involving the secant function. Conjecture. Let $p$ be an odd prime and define $$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
14
votes
1answer
562 views

Are fully general Frobenioids necessary?

Mochizuki's notion of a Frobenioid introduced in The geometry of Frobenioids I is rather elaborate. However, he also introduces a myriad of further properties that a Frobenioid may satisfy, and his ...
13
votes
1answer
316 views

Universality of $y^4-x^3$ mod $p$

For pedagogical reasons, I got interested in the equation $y^4-x^3=a$ over $\mathbf F_p$. To my surprise (maybe I'm naive), there is only one couple $(p,a)=(13,7)$ for which there is no solution, at ...
6
votes
2answers
508 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 ...
5
votes
1answer
288 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}...
6
votes
1answer
338 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....
-1
votes
2answers
127 views

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

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 $d \;| \; (2^ma + 2^nb)$?
0
votes
0answers
150 views
+50

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}-...
3
votes
0answers
62 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. ...
1
vote
1answer
82 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 ...
2
votes
0answers
59 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 ...
-1
votes
0answers
57 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 ...
-3
votes
0answers
73 views

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

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 ...
4
votes
2answers
211 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 $\ge n$, ...
5
votes
0answers
246 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
votes
1answer
170 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^...
6
votes
1answer
249 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 $...
0
votes
0answers
74 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\...
1
vote
2answers
178 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 ...
2
votes
0answers
81 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 ...
2
votes
1answer
80 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 $n$. ...
2
votes
0answers
110 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 ...
10
votes
0answers
193 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 \...
3
votes
0answers
203 views

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})\...
7
votes
0answers
182 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 ...
5
votes
1answer
87 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 ...
8
votes
1answer
215 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 ...
0
votes
0answers
108 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_{...
4
votes
0answers
245 views

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 ...
3
votes
0answers
87 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
votes
0answers
121 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
votes
3answers
552 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
votes
1answer
214 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\{...
0
votes
0answers
107 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 ...
4
votes
0answers
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-...