**2**

votes

**2**answers

99 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**0**

votes

**1**answer

92 views

### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...

**2**

votes

**0**answers

40 views

### On the uncountability of a subset of U-numbers of type $\leq m$

We say that $\xi\in \mathbb{R}$ is an $m$-ultra number if there exists a sequence $(\alpha_n)_n$ of $m$-degree real algebraic numbers, such that
$$
|\xi-\alpha_n|<(\exp^{[3]}(H(\alpha_n)))^{-n},\ ...

**2**

votes

**1**answer

116 views

### Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...

**1**

vote

**1**answer

172 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**0**

votes

**1**answer

174 views

### A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer ...

**2**

votes

**0**answers

67 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

**0**

votes

**2**answers

235 views

### Exponential Sum Bound

In
http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
...

**2**

votes

**1**answer

174 views

### Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*}
f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...

**0**

votes

**0**answers

187 views

### What does a Turing machine compute? [on hold]

I suspect that it might be necessary to define for a Turing machine how its inputs and outputs are to be interpreted in order to be able to say e.g. that a Turing machine $T$ computes an arithmetical ...

**1**

vote

**2**answers

219 views

### Rationale behind an requirement on Turing machines

Hopcroft and Ullman's definition of a Turing machine seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q_0,F \rangle$ obeying some ...

**0**

votes

**0**answers

59 views

### Must the radical of polynomial evaluated at integers be small enough at least once?

Basically I am interested if the radical of polynomial evaluated at integers can be small enough at least once.
Let $f \in \mathbb{Z}[x], \deg(f)>1$ be squarefree. For integer $a$ and $f(a) \ne 0$ ...

**0**

votes

**1**answer

134 views

### Composition of a transcendental function with a rational function [on hold]

The problem is: let $f: \mathbb{R}\to \mathbb{R}$ be an analytic transcendental function and let $\psi(x)=\frac{x}{2(1+x^2)}$. Is the function $f(\psi(x))$ transcendental?

**8**

votes

**1**answer

748 views

### What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...

**0**

votes

**1**answer

324 views

### Counterexample to Pólya's conjecture

It is known that Polya's conjecture is false and the smallest counter-example is about $10^9$.
Assuming that we are searching for a counter-example not knowing that it exists. What useful information ...

**1**

vote

**2**answers

227 views

### Asymptotics on prime divisors

Somewhat inspired from the Zsigmondy's theorem is my question. Suppose Let $a_{1}> a_{2}> \cdots > a_{k}$ be nonzero integers, with $k \geq 2$. Let $a(n) : = a_{1}^{n}+\cdots+a_{k}^{n}$ for ...

**-1**

votes

**0**answers

167 views

### Where is the Flaw in the Argument? [on hold]

Some days ago I have read about the Legendre Symbol. I found that there is no easy method for computation of the symbol. I tried to find out a method for determining the value of ...

**2**

votes

**1**answer

190 views

### A family Mersenne composite numbers?

I believe that the number
$$2^{2^{2t+1}+2t-1}-1$$
is composite for all positive integer $t$. I tested this for many $t$'s, but so far I didn't get a proof. Any idea?

**1**

vote

**0**answers

74 views

### On the computation of Asai L-function

I want so compute some simple twisted Asai L-function.
Let $E/F$ be a quadratic extemsion of number fields and $v$ a finite place of $F$.
Let $\chi$ be a unitary automorphic character of ...

**2**

votes

**0**answers

119 views

### Metric on the set of subsets of the rational primes

I was thinking how to say that two sets of prime numbers are close to each other, and came up with this.
NOTATION
$\ \Delta(A\ B)\ :=\ (A\setminus B)\cup(B\setminus A)\ \ $ is the symmetric ...

**9**

votes

**1**answer

221 views

### Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$:
$$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$
And here is a graphical representation of the 16-digit
"repetend," as a directed ...

**6**

votes

**1**answer

209 views

### Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...

**4**

votes

**1**answer

132 views

### Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& ...

**-1**

votes

**0**answers

105 views

### Primality matrices [on hold]

This question is some kind of a follow-up to my previous thread untitled About Goldbach's conjecture, the content of which follows: 'let's consider a composite natural number $n$ greater or equal ...

**9**

votes

**3**answers

938 views

### Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...

**0**

votes

**0**answers

171 views

### Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...

**-1**

votes

**0**answers

23 views

### Proof that $G(3)\le 7$ [migrated]

Let $G(k)$ be the minimal $n$ s.t. every sufficiently large integer is the sum of $n$ nonnegative $k$th powers.
Does anybody know where I can find Vaughan's proof that $G(3)\le 7$? I can't find a ...

**0**

votes

**0**answers

79 views

### Arthur's refinement of parameters for unitary automorphic representations

In his work on the classification of automorphic representations of a group $G$, Arthur has conjectured that the parameterization of such representations involves a homomorphism $\rho : SL_2 \times ...

**8**

votes

**1**answer

263 views

### Separation of lattice points on the Mordell elliptic curve

Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) ...

**0**

votes

**0**answers

86 views

### A System of Diophantine Equations [closed]

$p^2+1=2y^2$
$p+1=2x^2$
$p$ is prime and $x,y$ are integers. I conjecture that this only has solution for $p=7$

**2**

votes

**0**answers

38 views

### Extension of the product formula for valuations to a simultaneous completion

It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...

**1**

vote

**1**answer

143 views

### For a defined set $M$ (see problem) do there exist $a,b$ natural numbers so that $a,ab+1 \in M$

Let $\rho \in \mathbb{R}\setminus \mathbb{Q}$ be a irrational nuber, and
let $\varepsilon>0$ be arbitrarily small. Define $M=\{m \in \mathbb{N}: \exists k \in \mathbb{N}\hbox{ so that} |\rho m -k ...

**7**

votes

**1**answer

192 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...

**4**

votes

**0**answers

105 views

### Without Skolem–Mahler–Lech Theorem? [closed]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$.
Is there a "simple way" to prove this ...

**1**

vote

**1**answer

92 views

### On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$.
Let $\chi_v$ be a unramified unitary character of $F_v$.
Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...

**4**

votes

**0**answers

157 views

### Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)

Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...

**0**

votes

**0**answers

86 views

### Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.)
a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value.
consider ...

**5**

votes

**0**answers

108 views

### Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
...

**7**

votes

**5**answers

581 views

### Small values of a polynomial evaluated at roots of unity

The MO answer http://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant ...

**7**

votes

**1**answer

241 views

### The Diophantine equation $x^p - 4y^p = z^2$

If $p \geq 5$ is a prime, are there any integers $x, y, z > p$ such that
$(x, y) = 1$
and
$$x^{p} - 4y^{p} = z^{2}$$

**-1**

votes

**1**answer

124 views

### When is a local subring of a number field a valuation ring?

Do we have some good examples of local subrings of number fields which are not valuation rings?
Do we have an easy criterion for determining whether a local subring of a number field is a valuation ...

**1**

vote

**2**answers

304 views

### Practical use of estimates for the Gauss Circle Problem

This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) ...

**14**

votes

**2**answers

2k views

### Is there an algebraic number that cannot be expressed using only elementary functions?

(this is basically a repost of a question I asked at M.SE last year)
Is there an explicit real algebraic number (such that we can write its minimal polynomial and a rational isolating interval) that ...

**4**

votes

**1**answer

236 views

### Lower bound on the irrationality measure of $\pi$

There seems to be a lot of work on the upper bound for the irrationality measure of $\pi$, but I could not find anything on a lower bound except the general $\mu(\pi)\geq2$. Looks like it is the best ...

**3**

votes

**0**answers

234 views

### Is it possible to find explicit formula for the product $\prod_{d|n,\ d>1} (1-\mu(d)/\varphi(d))^{\varphi(d)}$?

I am trying to calculate the following product
$$
\prod_{d|n}_{d>1} \left( 1-\frac{\mu(d)}{\varphi(d)} \right)^{\varphi(d)}
$$
where the functions $\varphi$ and $\mu$ are Euler's totient and ...

**19**

votes

**1**answer

799 views

### A conjectured formula for Apéry numbers

A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in sci.math in 2005, but no proof was found.
Physicist Alan Sokal just reminded me of it, saying it was related to something he ...

**5**

votes

**0**answers

114 views

### Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and ...

**6**

votes

**1**answer

229 views

### Cohen-Lenstra heuristics for totally complex fields

If a number field $K$ is a Galois extension of $\mathbb{Q}$, and $G = \operatorname{Gal}(K/\mathbb{Q})$, then the class group of $K$ is a $\mathbb{Z}[G]$-module, and since $N = \sum_{g \in G} g$ acts ...

**5**

votes

**1**answer

213 views

### Kernel of the character of congruence groups

Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb Z)$ and $\chi:\Gamma \to \mathbb{C}^*$ a character of $\Gamma$ with finite image. Is then $\ker(\chi)$ also a congruence group? If not, can ...

**3**

votes

**1**answer

147 views

### Linear dependency of real numbers with integer coefficients adding up to zero [closed]

Let $x = (x_1, \dots, x_n)$ be a vector of real number. I was asking myself if there was an efficient way of telling whether there exists a non-zero vector of integers $z \in \mathbb Z$ such that both
...