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

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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, ...
2
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0answers
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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1answer
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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0answers
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])
3
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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 ...
3
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0answers
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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1answer
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 $\...
4
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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 ...
3
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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 ...
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2answers
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 ...
4
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1answer
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 $N$. ...
11
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1answer
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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1answer
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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1answer
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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1answer
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\\...
4
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0answers
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 ...
4
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4answers
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{\...
2
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0answers
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. ...
3
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0answers
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)=\...
2
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0answers
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{...
3
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0answers
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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1answer
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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1answer
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 $...
2
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1answer
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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0answers
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 ...
6
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0answers
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) ...
5
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0answers
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 ...
4
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1answer
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 $p_{n-1}<p_n< p_{n+1}$ ...
3
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0answers
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\...
3
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1answer
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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0answers
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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0answers
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 ...
1
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1answer
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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1answer
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 ...
0
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0answers
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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0answers
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 ...
-6
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0answers
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 ...
7
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0answers
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 $S\subset \mathbb{Z}$ ...
1
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1answer
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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1answer
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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0answers
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 ...
4
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0answers
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 ...
1
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0answers
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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0answers
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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0answers
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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0answers
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 $w=1$. ...

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