Prime numbers, diophantine equations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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2
votes
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
4answers
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
1answer
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
2answers
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
3answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
2answers
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
0answers
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
2answers
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
1answer
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
0answers
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
1answer
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 $k$ ...