**4**

votes

**1**answer

99 views

### Number of representations as sums of squares in rings of integers of number fields

Let $K$ be some number field, $\mathcal O_K$ denote its ring of integers, and let $n$ be a positive integer. Take $\alpha \in \mathcal O_K$, and consider the quantity $r_{n,K}(\alpha)$, which denotes ...

**19**

votes

**1**answer

780 views

### What were the main ideas and gaps in Miyaoka's attempted proof (1988) of Fermat's Last Theorem?

Out of sheer curiosity I have been reading Stewert and Tall's "Algebraic Number Theory and Fermat's Last Theorem" (2001). As it contains various bits of history, I found out to my own shame that I was ...

**1**

vote

**0**answers

93 views

### An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?

**1**

vote

**0**answers

120 views

### A curious limit [on hold]

Let $p$ be an odd prime number and let $a(n,p)$ be the denominator of $p^n/n$. Then it seems that
$$\lim_ ...

**5**

votes

**0**answers

84 views

### Intersection between the sums of the first integers, primes and non primes

Is the following conjecture true ?
$$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap
\left\lbrace ...

**1**

vote

**0**answers

89 views

### Possible counterexample to a conjecture of Granville about automorphisms of twists of hyperelliptic curves

This might be a counterexample to a conjecture of Granville
about automorphisms of twists of hyperelliptic curves.
In this paper,
the quadratic twist of $f(x)=y^2$ is denoted by
$C_d : d y^2=f(x)$ ...

**2**

votes

**6**answers

349 views

### Isotropic ternary forms

It is well known that some questions about isotropic ternary forms reduces to the study of the special case $f_0(X)=xz-y^2, X=(x,y,z)$, see page 301 of Cassel's "Rational quadratic forms" (Dover, ...

**-4**

votes

**1**answer

245 views

### Two conjectures in number theory [on hold]

Two conjectures in number theory
on July 01 2015, I read the Fermat last theorem, I propose two conjectures. But later Dr. Long Hai Dao let me that the first conjecture is the Lander, Parkin, and ...

**1**

vote

**0**answers

78 views

### Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions.
My question is whether it is known that if $a>4$
$$
\frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k}
...

**1**

vote

**1**answer

119 views

### Proofs needed for observations regarding prime-partitionable numbers

Below is the definition of a prime-partitionable integer taken from W. Holsztynski, R. F. E. Strube, Paths and circuits in finite groups, Discr. Math. 22 (1978) 263-272 and is apparently the same in ...

**23**

votes

**2**answers

2k views

### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...

**3**

votes

**0**answers

230 views

### Bounding the degrees in a Bézout relation for integer polynomials

Let $A$ and $B$ be two polynomials in $\mathbf Z[X]$ which generate $\mathbf Z[X]$, that is assume that there exist polynomials $U$ and $V$ in $\mathbf Z[X]$ such that
$$
A \cdot U + B \cdot V=1.
$$
...

**2**

votes

**0**answers

195 views

### Possible $\mathsf{NP}$ complete problem from number theory

A candidate $\mathsf{NP}$ complete variant of factoring was posted in http://cstheory.stackexchange.com/questions/4769/an-np-complete-variant-of-factoring, where decision problem ...

**26**

votes

**3**answers

1k views

### Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square.
$$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$
...

**2**

votes

**1**answer

153 views

### Minimum distance between factorials and powers of 2

Let's define for a positive integer $n$: $$a(n) = \min \{|n! - 2^m| : m \in \mathbb N \}.$$ Does there exist a good asymptotic lower bound for the values $a(n)$ for large $n$? In particular, is the ...

**7**

votes

**0**answers

157 views

### Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$
Is there a (geometrically irreducible) smooth variety ...

**3**

votes

**4**answers

432 views

### Integral points on a particular family of curves

This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that
$$
...

**8**

votes

**3**answers

262 views

### “Most Similar Vector Problem” on an Integer Lattice?

I am currently working on problem that I think could be expressed as an integer lattice problem.
Given $u \in \mathbb{R}^n$ and a bounded integer lattice $L = \mathbb{Z}^n \cap [-M,M]^n$ I would like ...

**7**

votes

**2**answers

171 views

### Random suborbits of a rotation

Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...

**0**

votes

**1**answer

198 views

### A particular interesting elliptic curve

Given the elliptic curve $E:y^2=x^3-4x+4$.
(a) How to prove that the group of rational points $E(\mathbb{Q})$ is generated by $P=(2,2)$.
(b) If we consider the piece of curve on the region ...

**44**

votes

**3**answers

6k views

### Is a “non-analytic” proof of Dirichlet's theorem on primes known or possible?

It is well-known that one can prove certain special cases of Dirichlet's theorem by exhibiting an integer polynomial $p(x)$ with the properties that the prime divisors of $\{ p(n) | n \in \mathbb{Z} ...

**2**

votes

**1**answer

193 views

### Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video.
From 33:00 to 37:00, it is said that after changing of variables, ...

**5**

votes

**1**answer

401 views

### Are there effective small intervals in which primes are dense?

As mentioned in Terry Tao's comment to this question, it is constructively known
that there are primes between sufficiently large cubes. $\:$ According to wikipedia,
"there exists a constant $\: ...

**7**

votes

**1**answer

261 views

### Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan

The following is a re-post from MSE because I did not get any answer even after offering a bounty.
Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern ...

**13**

votes

**2**answers

490 views

### What are Mean Values of Ideal Densities in Galois Extensions?

In an unfinished (and as of now unpublished) article intended for the encyclopedia of mathematics, Arnold Scholz wrote:
"Classifying extensions according to the Galois group
of their normal closure ...

**1**

vote

**1**answer

174 views

### reference on Dirichlet theorem on primes in arithmetic progression

I appreciate if you could help me to find a reference (and a proof).
Combining Dirichlet theorem on primes in arithmetic progression with Chebotarev densitiy theorem, we know that given two positive ...

**1**

vote

**1**answer

58 views

### Sampling from random unimodular matrices of a particular type?

Is there a nice way to parametrize unimodular matrices of form
$$\begin{bmatrix}
a1& a2& 0& 0\\
b1& b2& a1& a2\\
c1& c2& b1& b2\\
0& 0& c1& c2
...

**15**

votes

**3**answers

258 views

### Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$

$\newcommand{\Prb}[1]{\mathcal{P}_{#1}}$
I have the following number theory problem, related to Odlyzko's improvement on Shor’s factoring algorithm (see this cstheory.sx question for details).
Let ...

**0**

votes

**1**answer

274 views

### Particular case of Beal's Conjecture

Is it known that there exist no coprime positive integers $A$, $B$ and $C$ such that $A^3+B^4=C^3$? This is a particular case of Beal's Conjecture.

**1**

vote

**0**answers

92 views

### Finding a lower bound in terms of given integers

Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks

**26**

votes

**2**answers

2k views

### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
...

**4**

votes

**1**answer

481 views

### Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture.
We are looking for Weierstrass form and map from it of the genus one curve:
$$ y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0 $$
It is ...

**34**

votes

**1**answer

2k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**13**

votes

**1**answer

1k views

### Did Gauss know Dirichlet's class number formula in 1801?

Let $h_d$ be the number of $SL_{2}(\mathbb{Z})$ classes of primitive binary quadratic forms of discriminant $d$. It's natural to impose the hypothesis that $d$ is not at square, as we do below.
In ...

**7**

votes

**0**answers

126 views

### Modular factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K=\zeta\prod_\chi L(s,\chi)$$
with the Dirichlet characters distinct and ...

**8**

votes

**2**answers

586 views

### Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...

**16**

votes

**3**answers

2k views

### Fermat's proof for $x^3-y^2=2$

Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search,i found only proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's ...

**16**

votes

**3**answers

758 views

### Cases where the number field case and the function field (with positive characteristic) are different

In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the ...

**1**

vote

**0**answers

42 views

### Root number of an anticyclotomic twist

Let $\lambda$ be a self-dual Hecke character over a CM field $K$ with root number $-1$. How to show the existence of a finite order anticyclotomic Hecke character $\chi$ over $K$ such that the twist ...

**0**

votes

**1**answer

140 views

### Chinese remainder theorem for cyclic subfactor planar algebras

This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...

**3**

votes

**1**answer

78 views

### Any formula for the partial sum of a remainder series?

Let $N \ge 1$ be an integer, and there is a series $ \{ N \mod 1, N \mod 2, ... , N \mod i, ... \}$. Obviously when $i \gt N+1$, the series will become $\{N, N, N, ..., \}$. So only take $i \le N$ ...

**0**

votes

**1**answer

219 views

### Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?
Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...

**18**

votes

**5**answers

4k views

### number of zeroes in 100 factorial.

I was on math.stackexchange the other day and i found a question that said How many zeroes are there in 100!. I quickly factored it out and said that there where 24 zeroes. However thats only the ...

**11**

votes

**2**answers

309 views

### Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.
Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...

**3**

votes

**1**answer

875 views

### Does anyone have an electronic copy of Waldspurger's “Sur les coefﬁcients de Fourier des formes modulaires de poids demi-entier”?

Is there an electronic copy of Waldspurger's paper "Sur les coefﬁcients de Fourier des formes modulaires de poids demi-entier" floating around the internet somewhere? This appeared in J. Pures Math. ...

**2**

votes

**2**answers

143 views

### Divisibility among discriminants

Let $f(x)$ be an algebraic function over the field $\mathcal F$ of algebraic numbers over $\mathbb{Q}$. Suppose that $r \in \mathcal F$. Does the discriminant of $f(r)$ divide the discriminant of ...

**2**

votes

**1**answer

101 views

### degeneration of reductive group

If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...

**1**

vote

**0**answers

123 views

### Probability of correlated residues

Given $N,c\in\Bbb N$, where $c\ll(\log N)^{1/b}$ for any $b>1$ is fixed, what is the probability that given $A_1,A_2,A_3\in\Bbb N$ with ...

**2**

votes

**0**answers

101 views

### binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or
having a negative solution, arose from an ergodic theory question,
presumably itself currently intractible. I am not a number ...

**6**

votes

**1**answer

381 views

### Pure motives and compatible systems of $\ell$-adic representations

I am trying to understand the statement of the conjectures of Deligne on special values of certain $L$-functions, from his article titled, "Valuers de Fonctions L et periodes d'integrales" which ...