# Questions tagged [nt.number-theory]

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

12,961
questions

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6 views

### On the equations $(x^n+1)(y^n+1)=z^2+1$ and $(x^n-1)(y^n-1)=z^2+1$

Note that
$$(1^2+1)(2^2+1)=10=3^2+1\ \ \mbox{and}\ \ (1^4+2^4)(5^4+6^4)=8^4+13^4.$$
Today I tried to find positive integers $x,y,z$ satisfying $(x^4+1)(y^4+1)=z^4+1$ but failed. In view of this ...

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53 views

### Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series
$$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$
satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...

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113 views

### Small solutions of $f(x_1,…,x_n) \equiv 0 \pmod p$

Let $f(x_1,...,x_n)$ be polynomial with integer coefficients.
Is the following possible:
For almost all primes $p$ exist integers $X_i=X_1,...,X_n$ and $f(X_i)$ means $f(X_1,..,X_n)$.
such that:
$f(...

**2**

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**1**answer

192 views

### Does each prime $p>541$ have a quadratic residue $x^4+y^4<p$?

For any prime $p>5$, one of the numbers
$$1^2+1=2,\ \ 2^2+1=5,\ \ 3^2+1=10=2\times5$$
is a quadratic residue modulo $p$. In 2014 I conjectured that each prime $p$ has a primitive root $g<p$ of ...

**3**

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102 views

### Zagier's “From 3-manifold invariants to number theory”?

Zagier lectures on "From 3-manifold invariants to number theory" - do you know about texts of that or on the discussed web of ideas? ([https://www.mpim-bonn.mpg.de/de/node/10791])

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62 views

### The mystery of the jumps of functions with the prescribed jumps: Eisenstein series and hidden symmetries(?)

Say that a function $f(t)$ “changes only by jumps” if $f(t) + \text{const} = C ∑_k j_k θ(t-t_k)$ for a certain constant $C$. Here $θ(t)$ is the Heaviside
step function which has a jump 1 at $t=0$ (it ...

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133 views

### Is every Dedekind domain the integral closure of some principal ideal domain?

I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ ...

**2**

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**1**answer

91 views

### Watson's triple product for automorphic forms shifted by Maass rising operators

Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...

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78 views

### Origin of the term “singular integral” in the circle method

Ever since I learned about the circle method, I have implicitly held the following beliefs about the topic in the title:
The terms "singular integral" and "singular series" were ...

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105 views

### Reducibility of a cubic over a number field

Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible ...

**9**

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**2**answers

460 views

### Set of primes $p$ such that $\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset$

For which sets of primes $P$ is there a finite type $\mathbb{Z}$-algebra $A$ such that$$p\in P\iff\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset?$$Do all the finite $P$ arise this way?
$A=\mathbb{Z}/n$ works ...

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638 views

### Cubic function $\mathbb{Z}^2 \to \mathbb{Z}$ cannot be injective

It is easy to show, with an explicit construction, that a homogeneous cubic function $f: \mathbb{Z}^2 \to \mathbb{Z}$ is not injective. I am seeking a proof of the same result without the condition ...

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119 views

### Construction of genus class fields

Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...

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**1**answer

192 views

### An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$

I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
...

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**1**answer

360 views

### The Fibonacci sequence modulo $5^n$

Let $(F_k)_{k=0}^\infty$ be the classical Fibonacci sequence, defined by the recursive formula $F_{k+1}=F_k+F_{k-1}$ where $F_0=0$ and $F_1=1$.
For every $n\in\mathbb N$ let $\pi(n)$ be the smallest ...

**2**

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**1**answer

298 views

### Analytic continuation and convergence of a Riemann zeta related function

The functions in question are
$$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...

**16**

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**1**answer

552 views

### Possible contemporary improvement to bounded gaps between primes?

In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...

**3**

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**1**answer

118 views

### Finding a binary variable assignment to make a matrix with variables singular (over F_p)

Consider a square matrix defined over a finite field $M\in\mathbb{F}_p^{n\times n}$ having the following form
$$M=\begin{bmatrix}a_{11}+b_{11}x_1&a_{12}+b_{12}x_1&\dots&a_{1n}+b_{1n}x_1\\...

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53 views

### Distance function on generalized upper half planes

Let $\mathbb{H}^n$ be the quotient $GL_n(\mathbb{R})/O(n) \cdot \mathbb{R}^\times$, which at least in the theory of automorphic forms is called a generalized upper half plane. We could also think of $...

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53 views

### An invariant subspace under $G_Q$ action , and BSD-rank

Let $E/Q$ be an elliptic curve. $E(\bar{Q})$ is a complicated abelian group, which equals to all closed points of $\bar{E}$, and also, a $G_Q$-Galois module. Its torsion part, $E(\bar{Q})_{tor}$ is a ...

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**4**answers

423 views

### The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$

Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\...

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75 views

### Shimura curves and quaternion orders without elements of norm -1

Let $O$ be an order of a quaternion algebra over $\mathbb{Q}$ such that $O$ does not contain elements of norm $-1$. Such orders exist, but seems less used, in particular these orders are not Eichler.
...

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97 views

### Pythagorean triples and quadratic residues modulo primes

QUESTION. Are my following conjectures true? How to prove them?
Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that
$$\left(\frac ap\right)=\left(\frac bp\right)=\...

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49 views

### Simultaneous embeddings of ring of integers into product of rings

This question is about something mentioned in Katz's "$p$-adic $L$-functions for CM fields" in section 2.0.
Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and ring of integers $\mathcal{...

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79 views

### Norm $-1$ elements of quaternion algebras and Shimura curves [duplicate]

Let $Qa$ be an indefinite quaternion algebra over $\mathbb{Q}$.
Let $O$ be an order of $Qa$. The moduli space of abelian surfaces with quaternionic multiplication by $O$ is usually designed as the ...

**3**

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**1**answer

177 views

### Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...

**5**

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223 views

### Is $xz+1 $ a proper divisor of $a_3z^3+a_2z^2+a_1z+1$ finitely often?

Given a polynomial $P=a_3z^3+a_2z^2+a_1z+1, z >0$ with non-negative integer coefficients $a_1, a_2, a_3\ne 0$, it appears if $P$ is not factorizable then there are finitely many positive integers $...

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**1**answer

208 views

### Quadratic character of factorials

Let $p$ be a prime number and $S_p=\{(n!)^2 \bmod p, n=1,2,\dotsc,p-1\}$ the set of residues mod $p$ of squares of factorials. This set is obviously a subset of the group of quadratic residues mod p. ...

**3**

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285 views

### What is the definition of smoothness in scheme theory?

I want to ask if there is a somewhat desirable definition of "smoothness".
Definition. Let $k$ be a field and $X$ be a separated finite type scheme over $k$. Then $X$ is smooth if the ...

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110 views

### Is there an example of a non-principal Dirichlet character $\chi$ such that $\chi(F_n)\in \{0,1\}$ when evaluated at Fibonacci numbers $F_n$?

This is related to Sum of Fibonacci sequence evaluated at a Dirichlet character, but can be also be considered as a stand-alone.
I did an exhaustive search on non-principal (not necessarily primitive) ...

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86 views

### Are the nonnegative rationals diophantine with only two quantifiers?

Definition: A subset $D\subseteq \mathbb{Q}$ is diophantine if it is the projection of the zero set of a polynomial, i.e. there exists a polynomial $f\in\mathbb{Q}[X,Y_1,\dots,Y_n]$ for some $n$ such ...

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**1**answer

190 views

### Bertrand postulate- stronger version

Can we prove: let $2<p<q$ be two consequtive prime numbers, then there is always a prime number in interval $(q,p+q]$ (ie. betwen $q$ and $p+q$). In other words: if $p_{n-1}<p_n< p_{n+1}$ ...

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97 views

### Positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\{0,1,\ldots\}$ with $|x-y|>1$

I note that
$$2(n^2+n+1)^2 -1= n^4+(n+1)^4.$$
This leads me to pose the following question.
Question 1. Are there infinitely many positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\...

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**1**answer

115 views

### Centralizer of the absolute Galois group of a number field

By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...

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79 views

### How can I prove this statement assuming another statement? [closed]

Let
$$
X=\sum_{n=b+1}^∞ \frac{2^n(n!)^2+n+1}{(2n+1)!}
$$
where $b\in \mathbb{N}$ and $b\geq 13$.
Clearly, $X<\frac{1}{22}$ for all $b\geq 13$
Assuming that $X(2b+1)!$ will never be an integer, ...

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55 views

### On complexity of a particular prime problem

Is the following problem in $PH$ and is it complete for any class?
Problem: Is the $i$th bit of the $m$th prime $1$?
It appears to require a counting quantifier which has to demonstrate witness is the ...

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**1**answer

71 views

### Cyclotomic Numbers, Difference Sets

I have reading papers by Cunsheng Ding on Binary Cyclotomic Generators, Linear Complexity of Generalised Binary Sequences of Order 2. Since the topic is new to me understanding the text is quite ...

**2**

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**1**answer

144 views

### A family of difference sets (paper by A. L. Whiteman)

I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph, can ...

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71 views

### Strange lacunary Lambert series related to the Liouville function

Although I have my own interest for the Liouville function, I will suppress it here as the question seems to be interesting in its own right.
It occurred to me when I saw an answer by GH from MO to ...

**2**

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116 views

### Special zeta value and zeroes

Are there known relationships linking special values of the Riemann zeta function or MZV (multiple zeta values, i.e. $\zeta(n_1, \cdots n_k)$ with $n_i \in \mathbb Z^+$) to the nontrivial zeroes of ...

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129 views

### Sieve methods and primal height

Define the primal order $\omega_{\mathbb{P}}(n)$ of an integer $n$ as the smallest $k$ such that the $k$-th iterate of the prime counting function $\pi$ evaluated at $n$ is composite and the primal ...

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138 views

### Bounded Diophantine sets

A set $S\subset \mathbb{Z}$ is Diophantine if there is an integer polynomial $P(n, \bar{m})$ such that$$n\in S \iff (\exists \bar{m} \in \mathbb{Z}^{k})(P(n,\bar{m})=0).$$A set $S\subset \mathbb{Z}$ ...

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**1**answer

185 views

### On the relevance of the property $\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ a+b}(z)$ for the *fractional* iteration (“tetration”)

In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...

**10**

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**1**answer

870 views

### Normal numbers, Liouville function, and the Riemann Hypothesis

This is a question about whether or not some number $\lambda^*$ is normal in base 2. More specifically, I am wondering if $\lambda^*$ is not normal. Proving it is normal would be next to impossible, ...

**0**

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66 views

### What are some bounds on $\displaystyle \sum_{n\leq N} \mu(n)e(\alpha n)$? [duplicate]

As usual, let $e(x):=2\pi i x$. I already know of a bound on $\sum_{n\leq N}\mu(n)$: by the prime number theorem, it is $\ll x\exp(-c(\log x)^{\frac{1}{2}})$ for some $c>0$.
However, what happens ...

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93 views

### Riemann-Siegel formula for Dirichlet characters

After unearthing and giving a proof of what is now known as the Riemann--Siegel formula for the Riemann zeta function enabling the computation of $\zeta(1/2+iT)$ in time $O(T^{1/2})$,
in 1943 Siegel ...

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102 views

### On primes of specified length and bit pattern

Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...

**2**

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109 views

### On the connection between sums of prime numbers and distribution of prime numbers

As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between ...

**2**

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**0**answers

161 views

### David Applegate conjecture at OEIS sequence A237424 [closed]

The OEIS sequence is the sequence of the numbers of the form $$(10^a+10^b+1)/3$$
were $a$ and $b$ are nonnegative integers
Here is the link for the sequence https://oeis.org/A237424
This sequence has ...

**1**

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**0**answers

162 views

### Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...