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
0
votes
0answers
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 ...
7
votes
0answers
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
votes
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
votes
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
votes
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
votes
0answers
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
votes
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
votes
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
votes
0answers
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
votes
0answers
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
votes
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 ...
29
votes
0answers
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 ...
2
votes
0answers
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
votes
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
votes
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
votes
1answer
298 views
Analytic continuation and convergence of a Riemann zeta related function
The functions in question are
$$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...
16
votes
1answer
552 views
Possible contemporary improvement to bounded gaps between primes?
In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...
3
votes
1answer
118 views
Finding a binary variable assignment to make a matrix with variables singular (over F_p)
Consider a square matrix defined over a finite field $M\in\mathbb{F}_p^{n\times n}$ having the following form
$$M=\begin{bmatrix}a_{11}+b_{11}x_1&a_{12}+b_{12}x_1&\dots&a_{1n}+b_{1n}x_1\\...
4
votes
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 $...
0
votes
0answers
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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.$$
This leads me to pose the following question.
Question 1. Are there infinitely many positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\...
3
votes
1answer
115 views
Centralizer of the absolute Galois group of a number field
By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
-1
votes
0answers
79 views
How can I prove this statement assuming another statement? [closed]
Let
$$
X=\sum_{n=b+1}^∞ \frac{2^n(n!)^2+n+1}{(2n+1)!}
$$
where $b\in \mathbb{N}$ and $b\geq 13$.
Clearly, $X<\frac{1}{22}$ for all $b\geq 13$
Assuming that $X(2b+1)!$ will never be an integer, ...
0
votes
0answers
55 views
On complexity of a particular prime problem
Is the following problem in $PH$ and is it complete for any class?
Problem: Is the $i$th bit of the $m$th prime $1$?
It appears to require a counting quantifier which has to demonstrate witness is the ...
1
vote
1answer
71 views
Cyclotomic Numbers, Difference Sets
I have reading papers by Cunsheng Ding on Binary Cyclotomic Generators, Linear Complexity of Generalised Binary Sequences of Order 2. Since the topic is new to me understanding the text is quite ...
2
votes
1answer
144 views
A family of difference sets (paper by A. L. Whiteman)
I was reading A. L. Whiteman - A family of difference sets. On page 109, the author generalizes the reduced residue system modulo $v$ where $v=pq$, $p$ and $q$ are primes. At the last paragraph, can ...
0
votes
0answers
71 views
Strange lacunary Lambert series related to the Liouville function
Although I have my own interest for the Liouville function, I will suppress it here as the question seems to be interesting in its own right.
It occurred to me when I saw an answer by GH from MO to ...
2
votes
0answers
116 views
Special zeta value and zeroes
Are there known relationships linking special values of the Riemann zeta function or MZV (multiple zeta values, i.e. $\zeta(n_1, \cdots n_k)$ with $n_i \in \mathbb Z^+$) to the nontrivial zeroes of ...
-6
votes
0answers
129 views
Sieve methods and primal height
Define the primal order $\omega_{\mathbb{P}}(n)$ of an integer $n$ as the smallest $k$ such that the $k$-th iterate of the prime counting function $\pi$ evaluated at $n$ is composite and the primal ...
7
votes
0answers
138 views
Bounded Diophantine sets
A set $S\subset \mathbb{Z}$ is Diophantine if there is an integer polynomial $P(n, \bar{m})$ such that$$n\in S \iff (\exists \bar{m} \in \mathbb{Z}^{k})(P(n,\bar{m})=0).$$A set $S\subset \mathbb{Z}$ ...
1
vote
1answer
185 views
On the relevance of the property $\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ a+b}(z)$ for the *fractional* iteration (“tetration”)
In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \...
10
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
vote
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 $w=1$. ...
