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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

1
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0answers
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 ...
1
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0answers
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
votes
1answer
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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2answers
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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0answers
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}-...
0
votes
0answers
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 ...
3
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0answers
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. ...
1
vote
1answer
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 ...
2
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0answers
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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0answers
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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0answers
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 ...
4
votes
2answers
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 $\ge n$, ...
5
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0answers
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
votes
1answer
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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1answer
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 $...
0
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0answers
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\...
-1
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0answers
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
vote
2answers
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 ...
2
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0answers
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 ...
2
votes
1answer
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 $n$. ...
2
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0answers
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 ...
10
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0answers
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 \...
2
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0answers
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})\...
6
votes
0answers
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 ...
5
votes
1answer
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 ...
8
votes
1answer
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 ...
0
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0answers
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_{...
3
votes
0answers
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 ...
2
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0answers
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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0answers
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
votes
3answers
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
votes
1answer
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\{...
0
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0answers
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 ...
4
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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-...
2
votes
1answer
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}\...
3
votes
0answers
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}...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
3
votes
1answer
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 ...
2
votes
3answers
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?
4
votes
2answers
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
votes
2answers
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 \...
1
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0answers
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 ...
4
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0answers
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 ...
6
votes
0answers
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
votes
5answers
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 ...
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votes
1answer
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
votes
3answers
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
votes
1answer
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
votes
0answers
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 ...