# 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

10,454 questions

**10**

votes

**1**answer

194 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 ...

**-2**

votes

**0**answers

58 views

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

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 ...

**4**

votes

**0**answers

108 views

### Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?

For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...

**6**

votes

**2**answers

245 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 ...

**12**

votes

**2**answers

874 views

### Question on the 50th (known) Mersenne prime number

In a footnote to the list of known Mersenne prime numbers which can be found here, we read that the "ranking" therein is a provisional one since not all possible exponents between $37 \, 156 \, 667$ ...

**8**

votes

**0**answers

101 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 ...

**5**

votes

**1**answer

224 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}...

**-1**

votes

**0**answers

84 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}-...

**6**

votes

**1**answer

303 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....

**0**

votes

**0**answers

73 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\...

**2**

votes

**0**answers

79 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 ...

**1**

vote

**1**answer

77 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 ...

**-1**

votes

**2**answers

115 views

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

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 $d \;| \; (2^ma + 2^nb)$?

**3**

votes

**0**answers

54 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. ...

**2**

votes

**0**answers

53 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 ...

**0**

votes

**0**answers

51 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 ...

**4**

votes

**2**answers

199 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 $\ge n$, ...

**-3**

votes

**0**answers

62 views

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

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 ...

**5**

votes

**0**answers

242 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, ...

**5**

votes

**1**answer

237 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 $...

**2**

votes

**0**answers

180 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})\...

**2**

votes

**1**answer

160 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^...

**2**

votes

**3**answers

415 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?

**6**

votes

**1**answer

675 views

### Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that
$$\pi(x + y) - \pi(y) \leq \pi(x).$$
We can easily justify this heuristically, since
$$
\textrm{...

**13**

votes

**1**answer

1k views

### Elementary proof for Hilbert's irreducibility theorem

I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding.
I am only interested in the simple case where the ...

**20**

votes

**2**answers

2k views

### A Putnam problem with a twist

This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...

**0**

votes

**0**answers

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 ...

**1**

vote

**1**answer

138 views

### On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...

**10**

votes

**0**answers

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 \...

**-1**

votes

**0**answers

139 views

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

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=...

**3**

votes

**0**answers

231 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 ...

**1**

vote

**2**answers

175 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 ...

**5**

votes

**1**answer

311 views

### Convex subsets of sumsets

There's a famous old problem in additive number theory which asks whether, for any $\epsilon > 0$, if $B$ is a 2-basis for the set $A =$ {$1^2,2^2,...,n^2$}, i.e.: a set for which $B+B \supseteq A$,...

**0**

votes

**0**answers

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_{...

**5**

votes

**0**answers

629 views

### The Doi-Naganuma Lift

Let $F$=$\mathbb{Q}(\sqrt{D})$ be a real quadratic field
and $\mathcal{O}_F$ be the ring of integers of $F$.
The generating series $$\Omega^{(k)}(z_1, z_2, \tau ) := \sum^{\infty}_{m=1} m^{k-1} \omega^...

**5**

votes

**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 ...

**3**

votes

**1**answer

256 views

### Lower bound for k-fold Sidon Sets

k-fold Sidon set is defined in http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r25/pdf (page #4, paragraph 4)
Does anyone know what the best known lower bound construction is for the ...

**2**

votes

**1**answer

75 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

**0**answers

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 ...

**8**

votes

**0**answers

291 views

### Local properties of Galois representations attached to torsion classes

$\DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\GL}{GL} \newcommand{\F}{\mathbb{F}} \newcommand{\p}{\mathfrak{p}} \DeclareMathOperator{\Sym}{Sym}$
Let $F$ be a number field, and let $\Gamma$ be ...

**5**

votes

**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 ...

**3**

votes

**0**answers

86 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 ...

**6**

votes

**0**answers

172 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 ...

**8**

votes

**1**answer

211 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 ...

**25**

votes

**1**answer

1k views

### The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...

**3**

votes

**0**answers

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,\...

**7**

votes

**4**answers

657 views

### Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...

**5**

votes

**1**answer

210 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\{...

**9**

votes

**1**answer

454 views

### A question about Galois representations

Let $K$ be a number field and $(\rho,V)$, $(\rho',V')$ be two Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/K)$. Suppose that for some positive integer $n$ we have $\mathrm{Sym}^n\rho\...

**2**

votes

**1**answer

122 views

### Dimension formulae for Jacobi forms

I'm interested in locating dimension formulae for (more general) Jacobi forms associated with a lattice $L$ (where the Jacobi forms of Eichler-Zagier correspond to $L=A_1$).
Unfortunately, the ...