# Tagged Questions

**-1**

votes

**0**answers

14 views

### Integer solution for the equation EulerPhi(n^(n+1)) = 2^(2n+1)

Today, when I was investigating the Euler phi function, I noted that the integer solution n=4 satisfies the equation EulerPhi(n^(n+1)) = 2^(2n+1). It's the only I found, how to provide a proof?
...

**0**

votes

**0**answers

10 views

### A question on reduced, primitive, positive definite binary quadratic forms

Let $f(x,y) = ax^2 + 2bxy + cy^2$ be a positive definite binary quadratic form with even discriminant. We say that $f$ is primitive if $\gcd(a,2b,c) = 1$ and it is reduced if $0 < a < c$ and ...

**0**

votes

**1**answer

83 views

### Number of prime divisors of p^2-1 for a prime p

Let $n$ be an integer and $n=p_1^{a_1}\dots p_s^{a_s}$ be its factorization into primes. Denote by $\Omega(n)$ the sum of $a_i$. Does there exist a constant $k$ such that there are infinitely many ...

**1**

vote

**1**answer

47 views

### Relation of these two Dirichlet $L$-functions

Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet ...

**1**

vote

**1**answer

53 views

### Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for ...

**2**

votes

**2**answers

219 views

### How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...

**3**

votes

**1**answer

425 views

### On progress towards inverse Galois problem over rationals

I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.
From where I can read ...

**6**

votes

**0**answers

67 views

### Is the analogue of the Simple Continued Fraction in p-adic number fields useful?

Is there an analogue of the Simple Continued Fraction in p-adic number fields? Is it useful and does it have relations to best rational approximation in the p-adic sense? In the analytic case there is ...

**13**

votes

**3**answers

354 views

### Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the answer of Daniel Loughran. There is a ...

**6**

votes

**4**answers

704 views

### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...

**3**

votes

**0**answers

136 views

### Logarithmic bound for Diophantine equation

Let $a_1 \geq a_2 \geq a_3$ be given positive integers and let $N(a_1,a_2,a_3)$ be the number of solutions $(x_1,x_2,x_3)$ of the equation $$\dfrac{a_1}{x_1}+\dfrac{a_2}{x_2}+\dfrac{a_3}{x_3} = ...

**2**

votes

**1**answer

103 views

### Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...

**0**

votes

**0**answers

61 views

### Finding the number of integer points inside a sphere of radius R and dimension D centered at the Origin [on hold]

I am writing a computer program to count the number of integer points inside a sphere of radius R and Dimension D centered at the origin. In essence, if we have a sphere of dimension 2 (circle) and ...

**2**

votes

**1**answer

112 views

### Simultaneous $\pmod{p}$ congruences of two ternary quadratic forms

Let
$$\displaystyle f(x_1,x_2,x_3) = a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2,$$
$$\displaystyle g(x_1, x_2, x_3) = b_1 x_1^2 + b_2 x_2^2 + b_3 x_3^2$$
be two integral ternary quadratic forms with $f$ ...

**9**

votes

**1**answer

299 views

### Integral points on elliptic curves of the form $y^2=x^3+px$

As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be ...

**16**

votes

**2**answers

422 views

### The density of integers represented by a binary form

Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of ...

**3**

votes

**0**answers

156 views

### Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...

**3**

votes

**0**answers

173 views

### Number of solutions to $x_1x_2=x_3x_4\bmod n$

In https://www.math.ksu.edu/~cochrane/research/xyuvmodp.pdf it is shown $x_1x_2=x_3x_4\bmod p$ where $p$ is a prime has $\frac{|\mathcal B|}p+O(\sqrt{|\mathcal B|}\log^2p)$ solutions ...

**2**

votes

**0**answers

257 views

### An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m ...

**3**

votes

**1**answer

260 views

### Frobenius at ramified primes

Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T_p$ denote the $p$-adic Tate module of $E$, and let $V_p = T_p \otimes \mathbf{Q}_p$.
If the action of ...

**4**

votes

**0**answers

204 views

### Proof for new deterministic primality test possible?

Conjecture:
Let $n \in \mathbb{N}$ and $n$ odd.
Then the number $N=2^2 + n^2$ is prime, if and only if $N$ divides $2^{(N-1)/2} + 1$.
Thanks.

**5**

votes

**0**answers

85 views

### Primitive element for a number field, and ramification

Let $K=\mathbb Q(\theta)$ be a number field with integral primitive element $\theta$, and let $f(x)$ be the minimal polynomial of $\theta$. Let $p$ be a rational prime. It's well known that if $p$ ...

**1**

vote

**1**answer

199 views

### Finding cyclic subgroups of points on elliptic curves for isogeny based cryptography

Isogeny based cryptography is one of the newest post-quantum cryptography. Hardness of this system is based on finding isogeny between two elliptic curves. Also this is a theorem:
Elliptic curves ...

**2**

votes

**0**answers

284 views

+50

### Analytically continuing the limit of this series?

Main Question
I believe to following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times ...

**15**

votes

**1**answer

384 views

### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...

**3**

votes

**1**answer

180 views

### Non-vanishing of L-function of modular form

There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...

**6**

votes

**2**answers

316 views

### Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in ...

**1**

vote

**0**answers

95 views

### What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...

**3**

votes

**0**answers

74 views

### Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...

**4**

votes

**1**answer

199 views

### Reference for: Every local field can be realized as the completion of a global field

It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice ...

**5**

votes

**1**answer

325 views

### Unique quadratic subextension of a ray class field

Let $K_q$ denote the unique quadratic subextension of the ray class field over $\mathbb{Q}$ of conductor $q\times\infty$. Then $K_q$ should be $\mathbb{Q}(\sqrt{q})$ if $q$ if 1 mod 4 and ...

**8**

votes

**1**answer

190 views

### Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number ...

**11**

votes

**1**answer

591 views

### On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...

**1**

vote

**2**answers

291 views

### Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:
Dickson, L. E.. (1917). Fermat's Last ...

**17**

votes

**2**answers

807 views

### What is the smallest positive integer for which the congruent number problem is unsolved?

The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...

**3**

votes

**1**answer

120 views

### Irreducible monic polynomials

I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...

**4**

votes

**2**answers

188 views

### Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

Say we have a number field $K$. Let $G_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G_K$-module. We know that $H^1(K, M) := H^1(G_K, M)$, i.e. profinite group cohomology. For each place $v$ ...

**0**

votes

**0**answers

86 views

### Numbers with many prime divisors

Is there a positive $c$ such that for every $n$ there exists $m$ such that $2^m-1$ has at least $n$ distinct prime divisors and $m$ is not greater than $n^c$?
I'm also interested in this question ...

**2**

votes

**0**answers

67 views

### Is $\sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$ bounded as Im(s) goes to infinity?

Let $\rho$ denote the zeros of the Riemann zeta-function
and $\delta > 0$.
Is the function
$f(s) = \sum_{\rho} \frac{1}{(\rho - s)^{1 + \delta}}$
bounded as Im(s) goes to infinity?(the real part ...

**7**

votes

**1**answer

189 views

### Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$.
However, it is quite possible ...

**14**

votes

**1**answer

752 views

### What is the relationship between these two notions of “period”?

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...

**27**

votes

**3**answers

1k views

### $\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that
$M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$
and $\zeta(n)$, ...

**3**

votes

**3**answers

648 views

### Hypersurface missing just one point

Let $\mathbb F_q$ be a finite field and $n$ an integer.
What is the minimal degree $d = d(q,n)$ of a polynomial $f \in \mathbb F_q[X_1,\dots,X_n]$ such that the set $Z(f)$ of zeros of $f$ in the ...

**6**

votes

**1**answer

630 views

### A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number
So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...

**25**

votes

**6**answers

5k views

### How to find a closest integer point to intersection of two lines?

Here's a question that originates from StackOverflow (and the SO crowd isn't really qualified to solve it).
We're given two lines on the plane, specified by equations ($a x + b y = c$) whose ...

**1**

vote

**0**answers

197 views

### Is the difference of these two real-rooted functions real-rooted?

During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: ...

**2**

votes

**0**answers

74 views

### Conditions to check whether a prime ramifies in a certain way

Let $p$ be a prime number and let $f = a_n x^n + \cdots + a_0$ be an irreducible polynomial of degree $n \geq 3$ with a root $\alpha$. We say that $p$ ramifies in a powerful way over $K = ...

**8**

votes

**3**answers

739 views

### English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an ...

**-4**

votes

**0**answers

117 views

### A group theory problem? [closed]

Let $p$ be a prime s.t $p>2$ and k an integer s.t $k>1$, and l be a positive integer s.t $0<l<p^k<2^n$, does the following equation have any solutions in $Z_{2^n}$?
$$l^2=p^{2k}$$
In ...

**2**

votes

**0**answers

113 views

### Uniform bound on the Mordell-Weil rank of elliptic curves

I want to know that if there is an uniform upper bound for the rank of elliptic curves over $\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ...