**6**

votes

**4**answers

231 views

### Application and usage of representation of integers as sum of powers?

We know that there are many articles and manuscripts from the ancient to date talking about representation of integers as sum of squares, cubes etc. I would like to know what is it the usage and ...

**4**

votes

**1**answer

148 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 ...

**1**

vote

**0**answers

148 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 $7$ ?

**42**

votes

**1**answer

3k views

### What were the main ideas and gaps in Yoichi 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

114 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
$C_d : d y^2=f(x)$ ...

**6**

votes

**0**answers

162 views

### Intersection between the sums of the first positive 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 ...

**7**

votes

**3**answers

383 views

### Polynomials vanishing modulo some integer $n$

It is well-known that a polynomial $q \in \mathbb Z[t]$ vanishes modulo $p$ only if it lies in the ideal $J_p$ generated by $p$ and $t^p-t$. This means that either the degree is large (at least $p$) ...

**2**

votes

**1**answer

170 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

199 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 ...

**0**

votes

**1**answer

220 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 ...

**1**

vote

**0**answers

97 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}
...

**7**

votes

**2**answers

192 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 ...

**2**

votes

**0**answers

215 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 ...

**5**

votes

**1**answer

438 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 $\: ...

**1**

vote

**1**answer

189 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 ...

**8**

votes

**1**answer

320 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 ...

**2**

votes

**1**answer

100 views

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

Is there a way to parametrize totally unimodular $(3n+2)\times(2n+2)$ matrices of form
$$\begin{bmatrix}
\pm1 & \pm1 & 0 & 0 &\dots & 0 & 0 & 0 & 0\\
A_{2n} & ...

**1**

vote

**0**answers

100 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

**1**

vote

**1**answer

128 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 ...

**18**

votes

**3**answers

399 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

283 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.

**8**

votes

**3**answers

288 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 ...

**8**

votes

**0**answers

141 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 ...

**1**

vote

**0**answers

46 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 ...

**3**

votes

**1**answer

84 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 $i \le N$ ...

**2**

votes

**2**answers

146 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 ...

**4**

votes

**2**answers

198 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 ...

**0**

votes

**1**answer

226 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 ...

**2**

votes

**0**answers

123 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 ...

**1**

vote

**0**answers

149 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 ...

**4**

votes

**1**answer

487 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 ...

**2**

votes

**1**answer

196 views

### Counting function for prime pair with bounded gaps between them [duplicate]

I'll start by noting that I am not at all an expert on number theory. However I do use it in a proof and would like your assistance if possible.
Yitang Zhang breakthrough result established that ...

**2**

votes

**0**answers

161 views

### Is the twisted symmetric fifth power $L$-function holomorphic?

Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times ...

**6**

votes

**1**answer

398 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 ...

**0**

votes

**0**answers

82 views

### Finite-index subgroups of the ideles

Let $k$ be a number field and denote by $J_k$ the idele group of $k$. Recall that the finite-index open subgroups of $J_k$ which contain $k^*$ are very important in class field theory. My question ...

**13**

votes

**1**answer

574 views

### Elementary Proof of Infinitely many primes $\mathfrak{p} \in \mathbb{Z}[i]$ in the sector $\theta < \arg \mathfrak{p} <\phi $

A quick look at the primes in $\mathbb{Z}[i]$ suggests they might be evenly distributed by angle if we zoom out on a coarse enough scale.
I would like ask about the much weaker statement forgetting ...

**11**

votes

**2**answers

380 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 ...

**5**

votes

**0**answers

174 views

### Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...

**1**

vote

**1**answer

314 views

### Analytic Number Theory without Pigeonhole Principle [closed]

I don't know if this is an appropriate question for this website, but I will try my luck.
I am an undergraduate student, and recently I became interested in analytic number theory. When I started ...

**7**

votes

**1**answer

358 views

### What is the normal closure of $GL_2(\mathbb{Z})$ inside $GL_2(\mathbb{Z}_\ell)$?

This weird problem popped up in my research:
Let $\ell$ be a prime. Is there a description of the smallest normal subgroup of $GL_2(\mathbb{Z}_\ell)$ containing $GL_2(\mathbb{Z})$?
Is there a ...

**7**

votes

**0**answers

255 views

### Residue class sufficiency sets for the Collatz conjecture

I have recently managed to show a sequence of sufficiency sets for the Collatz conjecture whose natural density approaches 0 (the set theoretic limit approaches the set $\{1\}$). It is an extension of ...

**5**

votes

**1**answer

211 views

### Is $\liminf \frac{\sigma_{k}(n)}{n}$ finite for every $k$?

Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open?
Here, ...

**7**

votes

**0**answers

505 views

### “Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write:
In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...

**3**

votes

**4**answers

459 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
$$
...

**3**

votes

**1**answer

137 views

### Given n and q, how to find p so q$\neq$n-th power (mod p)?

Reasonable exceptions allowed on $q$. Example solution: $n=2$.
Suppose $q$ is odd. Let $p$ be so $pq\equiv -1$ (mod 8). Then $q\neq$ 2nd power (mod $p$) is the same as ...

**10**

votes

**1**answer

417 views

### When is the image of an integral polynomial contained in the image of another?

Suppose $f$ and $g$ are polynomials with integral coefficients and $f(\mathbb Z)\subset g(\mathbb Z)$. Is there any relation between $f$ and $g$?
For instance, this happens if $f=g\circ h$ for some ...

**0**

votes

**2**answers

242 views

### Intersection of two lattices

Suppose that $\Lambda_1, \Lambda_2$ are two sub-lattices of $\mathbb{Z}^n$ of full rank, defined by congruence modulo a prime $p$. That is, there exist two vectors with integer entries $\mathbf{a}, ...

**7**

votes

**2**answers

300 views

### Field of definition of dominant morphisms

Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes ...

**2**

votes

**0**answers

94 views

### Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...

**4**

votes

**1**answer

172 views

### Equidistribution of representations by a binary cubic form

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a binary cubic form with nonzero discriminant, and for a positive integer $m$ consider the integral representations $f(x,y)=m$. Assume that the number of ...