**3**

votes

**0**answers

255 views

### Fermat-Wiles “first case” in extensions of cyclotomic fields

I fell on the following fact : let p be an odd prime, let K denote the p-th cyclotomic field, let L be an extension of K with finite degree not divisible by p, and assume that the prime ideal $(1 - ...

**5**

votes

**1**answer

193 views

### The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ ...

**10**

votes

**0**answers

441 views

### Does Mochizuki's proof of abc conjecture gives an upper bound for the quality of a triple?

The quality of a triple $(a,b,c)$ of coprime positive integers with $a + b = c$ is defined as
$$q(a,b,c) := \frac{\log(c)}{\log(\mathrm{rad}(abc))}.$$
Then
$$a+b = c = ...

**0**

votes

**1**answer

798 views

### Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: ...

**11**

votes

**2**answers

365 views

### How many random sieve operations to decimate the set {2,…,n}?

Let $S$ be the set of integers $\{2,3,4,\ldots,n\}$.
Consider the following process:
Select a random element $k \in S$.
Remove from $S$ every number divisible by $k$.
Repeat with this reduced $S$.
...

**0**

votes

**0**answers

105 views

### The smallest disk containing all cirular arcs

In a comment to my recent question about covering segments by a disk, Gerhard Paseman has suggested a generalisation: replacing the segments of the original $n$-gon by a simple closed (say, convex) ...

**17**

votes

**1**answer

508 views

### Which philosophy for reductive groups?

I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the groupe $G$ we work on is to be ...

**-6**

votes

**0**answers

73 views

### Differential module for finite morphism

Let φ：A ---> B be an injective finite morphism between Artin rings over a field K. Especially φ is assumed to be injective.
Assume the degree of φ is equal to the fixed number d which is finite. That ...

**10**

votes

**2**answers

473 views

### Equation $x^2=y^p + 1$

can you help me please for solving this diophantine equation : $x^2=y^p+1$
and if you can give me a general method to studying such equation : $x^2=y^p+t$
Thanks

**1**

vote

**0**answers

87 views

### Elliptic curves with potential good reduction over a prescribed extension

Notation: Let $K/\mathbb{Q}$ be a quadratic number field; let $p\geq 3$ be a rational prime and let $\mathfrak{p}$ denote a prime lying above $p$; let $K_{\mathfrak{p}}$ denote the completion of $K$ ...

**2**

votes

**1**answer

112 views

### Families of ordinary Siegel Modular Forms

I'm looking for references to constructions and treatments of Hida Families/Eigenvarieties for ordinary Siegel modular forms (In particular: genus 2).
So far I've been reading Richard Taylor's thesis ...

**2**

votes

**1**answer

121 views

### How do the Dim($K_f / \mathbf{Q}$) vary for all f in a given $S_k(\Gamma(N))$?

For the theory of classical modular forms, the space of new forms $S_k^{new}(\Gamma(N))$ has a basis of Hecke eigenforms $\{ f_i = \sum a_n q^n : a_1=1, a_n \in \bar{\mathbb{Q}}\}$
Given $k$ and $N$ ...

**4**

votes

**1**answer

178 views

### Examples of perfect pseudo algebraically closed fields in positive characteristic

Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?

**0**

votes

**1**answer

106 views

### Relationship between quadratic residues modulo a prime and quadratic residues modulo a prime power [closed]

Been going through Alan Baker's A Comprehensive Course in Number Theory. Very interesting book, although the way proofs are presented sometimes throws me off a little.
I usually read through a ...

**4**

votes

**1**answer

272 views

### Show the upper bound of cardinality of $A$ is $C\sqrt{n\log{n}}$

$\forall l,m,n\in \Bbb{Z_+}$, let $A:=\{k: m+1\leq k\leq m+n\text{ and }l-k^2\text{ is a square number}\}$.
Please prove that the number of elements in $A$ is not more than $C\sqrt{n\log n}$, where ...

**1**

vote

**1**answer

131 views

### Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over ...

**0**

votes

**0**answers

120 views

### On odd perfect numbers $N = q{p^{2a}}{m^2}$ satisfying certain conditions

Let $N = q{p^{2a}}{m^2}$ be an odd perfect number, satisfying the conditions
$$\sigma(m^2) = p^{2a}$$
$$\sigma(p^{2a}) = q$$
and
$$q + 1 = 2{m^2}.$$
Note the following:
$p^a m < q$
$q$ is ...

**8**

votes

**0**answers

252 views

### Even Galois representations “mod p”

Consider an irreducible $\mathrm{mod}$ $p$ representation:
$$\rho: \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_2(\bar{\mathbb{F}}_p)$$
If $\rho$ is odd, it was conjectured by Serre in ...

**0**

votes

**0**answers

32 views

### Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...

**8**

votes

**1**answer

221 views

### Intuitive reasons for the existence of modular parametrizations

Whenever I encounter anything about modular parametrizations, I have a feeling it is something very unnatural: you have some kind of moduli space and all of a sudden it parametrizes an object ...

**4**

votes

**1**answer

187 views

### Motivation for cyclotomic units

I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?
...

**24**

votes

**3**answers

712 views

### Intuition for Zagier's theorem for $\zeta_K(2)$

In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v ...

**6**

votes

**1**answer

228 views

### Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$.
For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...

**2**

votes

**0**answers

118 views

### Galois conjugates of Hecke characters

Let $F$ be a totally real field. Let $K/F$ be a CM quadratic extension with CM type $\Sigma$. Let $E$ be the reflex field of $(K,\Sigma)$. For a Hecke character $\lambda$ over $K$ and $\sigma \in ...

**8**

votes

**0**answers

110 views

### Jacquet's approach to Rankin--Selberg L-functions

In his book "Automorphic Forms on GL(2), II", Springer Lecture Notes vol. 278, Jacquet defines the Rankin--Selberg L-function of $\pi_1 \times \pi_2$, where $\pi_i$ are automorphic representations of ...

**11**

votes

**1**answer

465 views

### Reference to a Don Zagier Result and the Congruent Number Problem

I was looking for a reference/explanation as to how Don Zagier managed to find the side lengths of a rational right triangle with area 157. There have been many literature references to the fact that ...

**1**

vote

**1**answer

164 views

### Radical of the sum $=$ radical of the product

My question is:
Has it been proved/disproved or studied the following?
For every $k\geq 4$ there are $k$ pairwise relatively prime numbers $a_1,a_2,...,a_k$ all greater than $1$ such that ...

**4**

votes

**1**answer

334 views

### On the mixed sum of three k-th powers

Let the set $S_k=\{\pm x^k \pm y^k \pm z^k \ \vert \ x,y,z \in \mathbb{Z} \}$.
Note that the signs are independently positive or negative.
First of all $S_2 = \mathbb{Z}$ because (see the answers ...

**-5**

votes

**1**answer

178 views

### How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? [closed]

If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ?
My feeling is that, this will be ...

**3**

votes

**1**answer

164 views

### regulators of number fields

Related question: Totally real number fields with bounded regulators
Given a number field $K$ with degree $n$ and determinant $D$, what is the "best" upper bound for its regulator $R$, if any? I know ...

**4**

votes

**0**answers

80 views

### The probability distribution of LCM of uniformly distributed integers in $\{1,\ldots,n\}$

In the recent paper by Fernandez and Fernandez here on ArXiv, the following formula which was first proved by Diaconis and Erdos appears, on page 2.
For $0<t\leq 1$ the distribution of the lcm of ...

**7**

votes

**2**answers

416 views

### Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here.
I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...

**6**

votes

**2**answers

331 views

### A different kind of divisor sums

Define, for a number $n$ the function $\mathcal{T}(n)$ which equals the number of different sums $d_1 + d_2,$ where $d_{1, 2}$ are divisors of $n.$ Now, one assumes that $\mathcal{T}(n)$ is ...

**14**

votes

**1**answer

648 views

### What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?

Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} ...

**-2**

votes

**0**answers

26 views

### Should my question be closed? [migrated]

See the question "On the number of $n$-perfect numbers". Some people claimed that it should be closed because I know the answer while I don't simply because they are hard open problems in number ...

**1**

vote

**1**answer

93 views

### Can a general binary sextic form be put into the following form?

Let $F(x,y) = a_6 x^6 + a_5 x^5 y + \cdots + a_0 y^6$ be a binary sextic form with real coefficients and non-zero discriminant. Can one always find an element $U = \begin{pmatrix} u_1 & u_2 \\ u_3 ...

**3**

votes

**2**answers

134 views

### What is the asymptotic growth rate of the product of divisor function up to n [duplicate]

This was asked in mathstackexchange (see here) but was not satisfactorily answered beyond my basic observations.
Let $\tau(k)$ be the number of divisors of the positive integer $k$.
How does ...

**5**

votes

**1**answer

554 views

### Why are integer points on elliptic curves interesting and useful?

I read some papers which dealed with integer points on elliptic curves. One of these papers are
http://projecteuclid.org/euclid.rmjm/1214947612.
My question is: Why are integer points on elliptic ...

**1**

vote

**1**answer

115 views

### Does this modification of the General Number Field Sieve factor integers?

The General Number Field Sieve
factors composite $n$ basically this way.
Select homogeneous polynomials with integer coefficients $f(x,y),g(x,y)$
s.t. $f(x,1),g(x,1)$ have common root modulo $n$ but ...

**1**

vote

**0**answers

79 views

### An apparent closed form for a slightly tweaked Dirichlet L-function. Could it be proven? [closed]

I made a small tweak to the well-known Dirichlet L-function ($p$=prime):
$$L(s, \chi_4) :=\prod_p \bigg(\frac {p^s}{p^s-\chi_4(p)} \bigg)=\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p ...

**0**

votes

**0**answers

25 views

### Beurling density $D(X)$ of $X=\{x_j\in\mathbb R, \ |x_j-x_{i}|>\gamma>0: \ i,j\in\mathbb Z\}$

Beurling density of set $X$ is defined (see, for example "Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces" by Aldroubi and Grochenig) as:
$$D(X)=\lim_{r\rightarrow \infty} ...

**4**

votes

**1**answer

167 views

### Transcendence of a ratio of p-adic logarithms

Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that ...

**6**

votes

**1**answer

250 views

### The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$
I am not really sure quite where to start here as I am ...

**2**

votes

**2**answers

124 views

### How large can a subset of $\{1,\ldots,N\}$ be if all pairwise LCMs of its elements are lower bounded?

Consider the set $\{1,2,\ldots,N\}$. Let $LCM(a,a')$ denote the lowest common multiple of the integers $a,a'.$
We say that $A\subset \{1,2,\ldots,N\}$ is $M-$good if $LCM(a,a')\geq M,$ for all ...

**3**

votes

**1**answer

285 views

### If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square?

Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help ...

**6**

votes

**2**answers

371 views

### Seeking an explanation for a peculiar factorization

Recently during my work, I encountered the following family of sextic polynomials
$$\displaystyle x^6 - 3x^5 + cx^4 + (5 - 2c)x^3 + cx^2 - 3x + 1.$$
By plugging in various values of $c$, I noticed ...

**10**

votes

**1**answer

574 views

### Relation between Weil Conjecture and Langlands Program

Recently I read Gelbart's An Elementary Introduction To The Langlands Program, which explained the origin of the program, and this question came to me. For an elliptic curve over finite field, the ...

**0**

votes

**2**answers

324 views

### For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$?

(Much revised for clarity.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
...

**0**

votes

**1**answer

143 views

### Is it possible to write identity for $ \{x(y^2-z^2)-y\}.\{u(v^2-w^2)-v)\}=a(b^2-c^2)-b$? [closed]

I asked this question in "math.stackexchange" but I did not get any response, so I put it here, maybe someone can help.
Is it possible to write identity similar to the identity
$$
...

**8**

votes

**0**answers

150 views

### Partition regularity in the squares

A linear equation $c_1x_1 + \cdots + c_sx_s = 0$ is partition regular if for every partition of the natural numbers into colour classes $A_1, \ldots, A_r$, there is a solution to the equation in which ...