**0**

votes

**0**answers

76 views

### All non-split Cartan subroups of $GL_2(\mathbb{Z}/n\mathbb{Z})$ are conjugate

Let $n>1$ be a positive integer and let $R$ be an order in an imaginary quadratic field with discriminant prime to $n$. Let $A=R/nR$ and let $\lbrace 1, \alpha \rbrace$ be a ...

**10**

votes

**0**answers

194 views

### Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that ...

**3**

votes

**0**answers

199 views

### Do those manifolds atrached to L-functions give rise naturally to motives?

Edited after Will Sawin's comment:
Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...

**10**

votes

**1**answer

301 views

### Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...

**6**

votes

**0**answers

266 views

### Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...

**3**

votes

**2**answers

111 views

### Density of polynomials which are soluble with respect to a set of primes

Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear ...

**5**

votes

**1**answer

255 views

### Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...

**1**

vote

**1**answer

98 views

### Counting primes powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is:
$$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...

**1**

vote

**1**answer

108 views

### Eisenstein series of weight $2$ for $\Gamma_0(N)$ : where am I wrong?

Let $A_{N,2}$ be the set of triples $(\psi,\varphi,t)$ such that $\psi$ and $\varphi$ are primitive Dirichlet characters modulo $u$ and $v$ with $(\psi\varphi)(-1)=1$, and $t$ is an integer such ...

**0**

votes

**0**answers

126 views

### A question on (odd) perfect numbers

I have asked this question in MSE a few weeks back, but did not receive any responses. I have cross-posted it to MO, hoping that it is appropriate for this site.
Let $\sigma(x)$ be the (classical) ...

**2**

votes

**1**answer

92 views

### Density of polynomials with a prescribed number field extension

For any polynomial $f(x) \in \mathbb{Z}[x]$, let $K_f$ denote the minimum splitting field over $\mathbb{Q}$ which contains all of the roots of $f$. Let $n \geq 2$ be a fixed integer, and let $K$ be a ...

**7**

votes

**2**answers

152 views

### approximate two different real numbers to order $\frac{1}{z^{3/2}}$

I took this result from Minkowski's book on Geometry of numbers:
Two arbitrary real quantitites $a$ and $b$ may be made to approach as near as we wish in value the two fractions $\frac{x}{z}$ and ...

**0**

votes

**0**answers

45 views

### Titchmarsh S function [on hold]

SO it is known that Titchmarsh S function $$ S(T)= \pi^{-1} arg\quad \zeta\bigg(\frac{1}{2}+iT\bigg)$$ under the assumption of riemann-hypothesis gives $$ S(T)=O(\frac{\log T}{\log \log T})$$ can ...

**7**

votes

**1**answer

209 views

### What are the strongest conjectured uniform versions of Serre's Open Image Theorem?

This question concerns the uniform conjectured effective versions and generalizations of these two results of Serre on $\ell$-adic Galois representations $\rho_{E,\ell}$ associated to a non-CM ...

**1**

vote

**0**answers

86 views

### Two spaces attached to mod 2 level 9 modular forms--a conjectural Hecke isomorphism

MOTIVATION
Nicolas and Serre have analyzed the structure of the space of mod $2$ modular forms of level $1$, viewed as a "Hecke-module". They show that for each $p>2$, the operator $T_p$ acting on ...

**5**

votes

**1**answer

104 views

### Bounding the number of lattice points inside an $n$-dimensional ellipsoid

I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie inside an $n$-dimensional ellipse.
That is, given an $n$-dimensional ellipsoid ...

**3**

votes

**4**answers

190 views

### Uniform upper bound for the sum over primes $\sum_{p \leq x} p^{-1+\varepsilon}$

I am reading the article D. M. Gordon and C. Pomerance, The distribution of Lucas and elliptic pseudoprime, Math. Comp. (1991) (click).
In equation (27) the authors, apparently, used the following ...

**0**

votes

**0**answers

22 views

### Degree $2$ nilpotent matrices with non-zero product [migrated]

Let $n$ be sufficiently large positive integer.
Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$
or $\mathbb{Z}/n \mathbb{Z}$.
Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and
...

**5**

votes

**1**answer

168 views

### Size of smallest multiple $\sum \epsilon_i p^i$ of $n$ with digits $\epsilon_i\in \{0,1\}$ for $p$ a prime

Any natural number $n$ coprime with a prime number $p$ is a divisor of
$M=1+p+p^2+\dots+P^{N-1}$ where $N\leq \varphi((p-1)n)$ is the order of
$p$ in $\left(\mathbb Z/((p-1)n)\mathbb Z\right)^*$.
...

**10**

votes

**1**answer

351 views

### Class field towers

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...

**-1**

votes

**0**answers

81 views

### X^2 + 4 = y^5 is there any solution to this [on hold]

is there any integer solution to the given equation?
i have tried to solve the problem considering modulo of different prime but it just aint working
it'd be nice if i get a solution soon

**7**

votes

**1**answer

194 views

### Asymptotic limit of truncated Legendre sieve

Consider the truncated sum
$$
S(x):=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}\mu(d)/d,
$$
where $P(z)$ is the product of all primes less than or equal to $z$, and $\mu(d)$ is the Möbius ...

**4**

votes

**1**answer

272 views

### Finite field “contour” sum

Let $\Bbb{F}_q$ be a finite field. Choose a non-square $\delta \in \Bbb{F}_q^*$
and form the quadratic extension $\Bbb{F}_q\big( \sqrt{\delta} \, \big)$. For
an element $z \in \Bbb{F}_q\big( ...

**3**

votes

**0**answers

278 views

### Explicit Galois Action for $X^3 - X -1$ [migrated]

I have always been frustrated with how indirect discussions of Galois Theory are in Algebra textbooks. Even in fine treatments such as Miles Reid.
Are there any good examples where we can draw ...

**7**

votes

**1**answer

315 views

### Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...

**17**

votes

**3**answers

1k views

### A Polynomial With Positive Prime Density

Let $P(x)$ be a non-constant polynomial with real coefficients.
Can natural density of
$$\{n\ |\ \lfloor P(n)\rfloor \ \text{is prime.}\}$$
be positive?

**2**

votes

**0**answers

130 views

### Squarefree part of a Mersenne number

Consider the Mersenne number; $M_p=2^p−1$.
Let $M_p=a_pb^2_p$ where $a_p$ is positive, squarefree, and $p$ is prime.
A chinese paper written by Le Maohua "“On Mersenne Numbers”" states that the ...

**-1**

votes

**0**answers

110 views

### Groupes fondamentaux de Tate mixte [closed]

Have anyone ever read deligne and Goncharov's paper,Could you give me some idea why the formula(5.16.1) is true?this paper is very easy to find on the internet.So please forgive me not giving the ...

**6**

votes

**3**answers

253 views

### Nontrivial solutions for $\sum x_i = \sum x_i^3 = 0$

For $x_i \in \mathbb{Z}$, let $\{x_i\}$ be a fundamental solution to the equations:
$$
\sum_{i= 1}^N x_i = \sum_{i=1}^N x_i^3 = 0
$$
if $x \in \{x_i\} \Rightarrow -x \notin \{x_i\}$.
For instance, a ...

**7**

votes

**0**answers

165 views

### Integer solutions of $x^2=4+8y^2+13z^2$

I have been looking for integer solutions of certain Diophantine equations, one of the simplest examples being
$x^2=4+8y^2+13z^2$.
The ideal answer would be a way to parametrize all the integer ...

**3**

votes

**1**answer

108 views

### Atkin-Lehner theory for nonholomorphic Eisenstein series

I am currently reading something about nonholomorphic Eisenstein series $E_\mathfrak{a}(z,1/2+it)$ for $\Gamma_0(q)$, where $\mathfrak{a}$ is a cusp (cf. Iwaniec, H. Spectral Methods of Automorphic ...

**15**

votes

**1**answer

813 views

### Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of
$$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...

**3**

votes

**1**answer

150 views

### Hodge-Tate weights of induced representation

Let $K$ be a finite extension of the field of $p$-adic numbers $\mathbb{Q}_p$ and let $E$ be another such extension, such that all the $\mathbb{Q}_p$ embeddings $K \to \bar{\mathbb{Q}}_p$ are ...

**6**

votes

**1**answer

277 views

### Primes isolated by large gaps to either side

Say that the $n$-th prime $p_n$ is isolated to degree $k$
(my notation) if
the prime gap to either side is larger than $\log p_n$ to the $k$-th power:
\begin{eqnarray*}
p_n - p_{n-1} & > & ...

**7**

votes

**1**answer

278 views

### What's the difference between Euler systems and Kolyvagin systems?

Is there a difference between Euler systems and Kolyvagin systems - or do they refer to the same thing? For example there is the Heegner point Euler system, but you don't really see a Heegner point ...

**-2**

votes

**0**answers

64 views

### Integer solution to the equation [migrated]

Does there exists an integer solution (for every integer $m\geq 1$) for the following equation?
$$x_1x_2...x_n+(2y+1)z+y=4m+3$$
where, $1\leq x_1\leq x_2\leq...\leq x_n\leq l$,$0\leq y \leq ...

**0**

votes

**1**answer

284 views

### Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ ...

**3**

votes

**0**answers

48 views

### Noncommutivity of various lifts

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ ramified at $p$ and $\infty$ for simplicity. Then an automorphic form on $D^{x}/F^{x}$ has several different interpretations. First, through ...

**2**

votes

**2**answers

325 views

### Primes as uncorrelated random variables [closed]

The heuristic justification section of the Wikipedia article about Goldbach's conjecture says that the argument that suggests that
the number of twin primes below $x$ should be roughly ...

**3**

votes

**4**answers

310 views

### Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root

Are there examples of polynomials $x_1(t), x_2(t) \in \mathbb{Q}[t]$ of equal degree at least one, with $\gcd(x_1(t), x_2(t)) = 1$, such that the sum $(x_1(t))^4 + (x_2(t))^4$ is divisible by the ...

**5**

votes

**1**answer

164 views

### An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let
$$
\sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}]
$$
be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$.
...

**4**

votes

**1**answer

140 views

### Sets of natural numbers such that sums of a bounded number of its elements form a semigroup

This is a naive question and I'm afraid it might be better placed on math.se. I would like to leave it to your judgement.
I would like to know what is known about sets $A$ of natural numbers such ...

**47**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**13**

votes

**1**answer

438 views

### When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...

**14**

votes

**3**answers

2k views

### A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known.
Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...

**11**

votes

**2**answers

455 views

### distribution of $\sqrt{-1} \mod p$

While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$.
Let $r_p$ be an integer with $0\leq r_p < p$ and $r_p^2 \equiv ...

**-2**

votes

**0**answers

45 views

### how to calculate the radius of convergence of the p-exponentials of Pulita?

please it is known from the Pulita thesis that the radius of convergence of his pi-exponentials is 1; he used a differential operator which has this pi-exponetial as a solution etc..., but me I ...

**4**

votes

**2**answers

590 views

### Is every positive integer a sum of at most 4 distinct quarter-squares?

There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference?
Examples:
quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$
2-term sums: ${2+1, 4+1, ...

**15**

votes

**1**answer

531 views

### Question on the irrationality of $e$

I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...

**7**

votes

**1**answer

210 views

### how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...