# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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, ...
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### 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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### 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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### 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 $n>1$...
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### 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 ...
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 ...
### 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 $w=1$. ...