**2**

votes

**1**answer

341 views

### Can the Units of a Cubic Field be Proven from Pigeonhole Principle alone?

I would like to run through the proof of Dirichlet Unit Theorem for a cubic field.
Let's try $\mathbb{Q}[x]/(x^3 - x - 1)$. This has 1 real root and 2 complex roots (or embeddings).
The units in ...

**1**

vote

**1**answer

203 views

### Quadratic squares

Consider a quadratic polynomial $p(x) \in \mathbb{Z}[x],$ say
$$p(x) = a x^2 + b x + c.$$
The question is: is there an asymptotic estimate for the number of integral $x$ in $[-N, N]$ for which $p(x)$ ...

**3**

votes

**1**answer

186 views

### Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...

**2**

votes

**1**answer

280 views

### Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?

Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$.
Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...

**1**

vote

**0**answers

216 views

### Does $8a+5$ ever divide $b^2+8$? [migrated]

For natural $a,b$, does $8a+5$ ever divide $b^2+8$ ?
It doesn't for $b$ up to $10^7$.
Couldn't find congruence obstructions for moduli up to $500$.
$b^2+8$ can be even.

**0**

votes

**0**answers

8 views

### Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? [migrated]

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle?
In other words, could any of you give me a algebraic proof for the following?
given prime ...

**6**

votes

**1**answer

221 views

### Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...

**7**

votes

**0**answers

262 views

### Capitulation of ideal classes in general Dedekind Domains

I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up:
Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...

**29**

votes

**1**answer

3k views

### Is the following identity true?

Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ ...

**2**

votes

**0**answers

84 views

### $U_p$ operator is not compact on $p$-adic modular forms

I know that one of the reasons for introducting overconvergent $p$-adic modular forms is that the $U_p$ operator is compact on them.
Is there an easy way to see that $U_p$ is not compact on non-...

**2**

votes

**0**answers

96 views

### q-expansion principle for Γ(N)

I would like to know if there's a $q$-expansion principle for $\Gamma(N)$.
Namely, let $f$ be a weight $k$ weakly holomorphic modular form for $\Gamma(N)$ whose $q$-expansion at infinity has ...

**1**

vote

**1**answer

117 views

### Level dependence in the Ramanujan-Petersson Conjecture for GL(2) Maass forms

Suppose $f(z) = \sum_{n \geq 1} A(n)n^{\frac{k-1}{2}} e(nz)$ is a weight $k$ holomorphic cusp form on $\text{GL}(2)$. Then the Ramanujan-Petersson conjecture (proved in this case by Deligne) says ...

**4**

votes

**1**answer

234 views

### Imprimitive solutions to $x^2+y^3=z^7$

Poonen, Schaefer, & Stoll give the primitive solutions to $x^2+y^3=z^7$:
$$
(±1, −1, 0), (±1, 0, 1), ±(0, 1, 1), (±3, −2, 1), (±71, −17, 2),\\
(±2213459, 1414, 65), (±15312283, 9262, 113), (±...

**0**

votes

**1**answer

70 views

### least integer for a related factorial

Let $p$ be a prime, $d$ a divisor of $p-1$, $j$ an integer with $0 \le j<d$. Let $n\in\mathbb N$. Does there exist a formula giving the least integer $m$ such that
$$v_p((dm)!)\ge v_p((dn)!)+j+dn?$$...

**0**

votes

**2**answers

161 views

### Certain totient chain growth assuming Carmichael's Totient Function Conjecture

Assume the Carmichael's Totient Function Conjecture.
Consider the totient chain
$$n_0=\phi(3^2)\rightarrow n_1=\phi((\phi^{-1}(n_0))^2)\rightarrow n_2=\phi((\phi^{-1}(n_1))^2)\rightarrow\dots$$
...

**8**

votes

**1**answer

247 views

### Sequence of nested sets in $[0, 1]$ with bound on gaps

What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...

**7**

votes

**1**answer

325 views

### Quasi-split tori and algebraic groups

Let $k$ be a perfect field.
Recall that an algebraic torus $T$ over $k$ is called quasi-split if there exists some finite étale $k$-algebra $A$ such that
$$T \cong \mathrm{R}_{A/k} \mathbb{G}_m.$$
A ...

**2**

votes

**1**answer

221 views

### Non-negative integer solutions of x^2+y^3=n

I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How ...

**3**

votes

**1**answer

177 views

### Tate modules of elliptic curves with complex multiplications

Let $E/K$ be an elliptic curve with complex multiplication
over an imaginary quadratic field $K$. Then, I heard that
it is well-known that the Tate module $V_{p}(E)$ over
$\mathbb{Q}_{p}$ ...

**8**

votes

**1**answer

222 views

### Imaginary quadratic fields: Euclidean if and only if norm Euclidean

Let $K$ be an imaginary quadratic field and $O_K$ be its ring of integers. We say $O_K$ is norm Euclidean if the norm is a Euclidean function. It is known from the classification of imaginary ...

**0**

votes

**1**answer

114 views

### Products of varieties of index 1

Let $k$ be a field of characteristic 0 and let $X$ and $Y$ be smooth, projective and geometrically integral $k$-schemes of finite type. Assume that both $X$ and $Y$ have 0-cycles of degree 1. Does $X\...

**0**

votes

**0**answers

83 views

### Montgomery's pair correlation conjecture and FRFT [on hold]

The Wikipedia article about Montgomery's celebrated pair correlation conjecture says it was investigated by its author through Fourier transform, but has the fractional Fourier transform $\mathcal{F}_{...

**21**

votes

**4**answers

1k views

### Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ?
Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...

**8**

votes

**1**answer

209 views

### $p | f(x)$ if and only if $p^k | x$.

Given a prime number $p$ and a positive integer $k$. Consider integer-valued polynomials $f$ satisfying the property that $p | f(x) \Leftrightarrow p^k | x$.
Question. What is the smallest degree of ...

**9**

votes

**2**answers

464 views

### Dissecting Ramanujan´s Cuboid: 1729 = 19 x 13 x 7

Consider the cuboid of dimensions 19 x 13 x 7 whose volume is 1729, the Hardy-Ramanujan number. What is the least number of smaller cuboids into which it can be dissected so that the resulting pieces ...

**30**

votes

**3**answers

2k views

### Is there a nice explanation for this curious fact about cyclic subgroups?

Here's something that I noticed that quite surprised me.
Let $G$ be a finite abelian group. Consider the following expression.
$$
\nu(G) = \sum_{\substack{H \leq G \\ H \text{ is cyclic}}} |H|
$$
It ...

**4**

votes

**0**answers

85 views

### In what range can we find diophantine approximations using the LLL-algorithm?

Let $\alpha_1, \ldots, \alpha_n$ be $\mathbb{Q}$-linearly independent real numbers. I want to show that for all $x_1, \ldots, x_n\in\mathbb{Z}$, $|x_i|<N$ we have some lower bound for $\left|\sum ...

**2**

votes

**1**answer

148 views

### Generating function for numbers divisible by some primes

Consider the first $k$ primes $p_1 = 2, p_2 = 3, \dots, p_k$. Let $A_k$ be the set of numbers that are divisible by at least one $p_i$. We can represent this set as a generating function:
$$G_k(x) = \...

**0**

votes

**0**answers

101 views

### Normalizer of non-split tori

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$.
Question: What do we know about the normalizer $N_G(T)$...

**0**

votes

**0**answers

55 views

### Is there a generalized Mac-Lauren summation formula for this sum?

Let $\chi$ be a Dichlet character and $f$ be a derivable function $k+1$ times. Is there a generalized Mac-Lauren summation formula for this sum $\sum_{n \leq x}\chi(n) f(n)$ i.e is there an ...

**1**

vote

**0**answers

144 views

### Researching the irrationality of a number [closed]

I am conducting a little research on checking if a number, written in positional numeral system is irrational.
Let $h^p_n$ be the most right non-zero digit of number $n!$ written in numeral system ...

**2**

votes

**0**answers

54 views

### Prescribed norm residue symbol in number field

Suppose $F$ is a number field, and $a, b$ are non-zero elements. Does there always exist $x \in F$ such that the norm residue symbols (=cup products) are $(a, x)= 0 = (x, b) \in H^2(F, \mathbb{F}_2)$ ...

**2**

votes

**0**answers

186 views

### Density of set of unique sums

Set $U_1 = \{1,2\}$, and for $n\in\mathbb{N}$ with $n\geq 1$ we set $U_{n+1} = U_n\cup A_n$ where $A_n$ is the set of elements of $\mathbb{N}$ that are not in $U_n$ but can be uniquely written as a ...

**7**

votes

**1**answer

293 views

### Groupoid cardinality and Egyptian fraction representations of 1

It is well-known that any rational number can be represented using a sum of distinct Egyptian fractions (that is, rational fractions of the form $1/n$ with $n\in\mathbb{N}$). This may be proven by ...

**0**

votes

**1**answer

156 views

### How many the distinct linear factors of $f(x)-f(y)$ can be for f in Q[x]?

Let $f \in \mathbb{Q}[x]$.
Let $S(f)$ denote the number of distinct linear factors
of $f(x)-f(y)$.
$S(f)$ is bounded by $\deg(f)$.
Q1 Is $S(f)$ bounded by constant?
Q2 Is it possible $S(f)&...

**8**

votes

**0**answers

258 views

### What is an example of a non-mixed $\ell$-adic sheaf?

$\def\FF{\mathbb{F}}\def\cG{\mathcal{G}}\def\QQ{\mathbb{Q}}\def\CC{\mathbb{C}}$I've been attending a reading seminar at Michigan on Kiehl and Weissauer's book Weil conjectures, perverse sheaves and l’...

**0**

votes

**1**answer

220 views

### On the quadratic reciprocity law? [closed]

In the Quadratic Reciprocity Law
$$\exists x\in\Bbb{N}\quad x^2\equiv p\pmod q\iff\exists y\in\Bbb{N}\quad y^2\equiv q\pmod p$$ if $p\equiv q\equiv 1\pmod4$.
Is there any relation between $x$ and $y$ ...

**4**

votes

**0**answers

214 views

### étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$.
Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$.
For ...

**7**

votes

**1**answer

174 views

### Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$.
Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1].
But perhaps that estimate is too crude. ...

**2**

votes

**1**answer

93 views

### Irreducible binary quartic form with prescribed $I$ and $J$-invariants

Let $F(x,y) = ax^4 + bx^3y + cx^2y^2 + dxy^3 + ey^4$ be a binary quartic form with integer coefficients. It is well-known that $F$ has two algebraically independent invariants under the action of $\...

**2**

votes

**3**answers

380 views

### Learning roadmap for algebraic number theory

I have read some elementary number theory from David Burton's text and I know groups and rings from Herstein's book Topics in Algebra and some field theory from different sources online. I am ...

**5**

votes

**0**answers

87 views

### What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...

**1**

vote

**0**answers

58 views

### A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and
$$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...

**4**

votes

**2**answers

270 views

### Breaking the RSA encryption based on a $(e,N)$ given an integer $w \neq 0$ such that $e^w = 1 \mod(N)$?

In his book 'Forcing with Random Variables and Proof Complexity' Jan Krajíček claims (p.154) that it is possible to break the RSA encryption with public key $(e,N)$ if one has has an integer $w \neq ...

**7**

votes

**0**answers

184 views

### horocycle flow and the prime number theorem

Looking at Zagier's Eisenstein Series and the Riemann Zeta Function, we get a proof of the prime number theorem using horocycles. I would really love it if there were a geometric proof like this.
...

**2**

votes

**2**answers

187 views

### Lower bound for the number of representations of integers as sum of squares

Let $k\geq 4$. As usual, let $r_k(n)$ denote the number of ways to represent $n$ as the sum of $k$ squares. Is this true that for every $\varepsilon>0$, one has $r_k(n) \gg n^{\frac{k}{2}-1-\...

**1**

vote

**1**answer

466 views

### What is the best currently proven bounds on prime gaps?

I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best ...

**6**

votes

**0**answers

213 views

### Universal Property of Fontaine's Period Ring $B_{dR}^+$

In the introduction to his Asterisque Expose "Le Corps des Periodes p-Adiques",
Fontaine announces a characterization of $B_{dR}^+$ by some universal property. Unfortunatly,
at least for $B_{dR}^+$ ...

**1**

vote

**1**answer

153 views

### s(n) = kn or s(n) = n/k? [closed]

This is not an important question, just for fun.
Definition:
$\sigma (n)$ = sum of the positive divisors of $n$.
$s(n)$ = sum of the proper positive divisors of $n$.
For $s(n) = kn$ , where $k$ ...