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        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

        1
        vote
        0answers
        11 views

        The arithmetic meaning of opers (if any)

        Let $G$ be a complex, connected semi-simple Lie group, $G'$ its Langlands dual group, $\mathrm{Bun}_G$ the moduli stack of $G$-bundles on a smooth projective curve $\Sigma$ over complex numbers, $\...
        3
        votes
        1answer
        43 views

        $\sum_{i=1}^x\sum_{j=1}^xf(i\cdot j)$ Double Summing a (Not Completely) Multiplicative Function

        Let $f(n)$ be a multiplicative function that is not completely multiplicative, i.e $f(m)\cdot f(n)= f(m\cdot n)$ only if $gcd(m,n)=1$. Let $S(x)$ be the double sum over $f$, that is: $$S(x)=\sum_{i=1}...
        6
        votes
        0answers
        123 views

        Greatest prime factor of n and n+1

        For a positive integer $n$ we denote its largest prime factor by $\operatorname{gpf}(n)$. Let's call a pair of distinct primes $(p,q)$ $\textbf{nice}$ if there are no natural numbers $n$ such that $\...
        8
        votes
        0answers
        86 views

        Number rings with elements of norm $-1$

        Is there a nice characterization of number rings $\mathcal{O}$ such that there exists an element $x \in \mathcal{O}$ whose norm is $-1$? One obvious necessary condition is that $\mathcal{O}$ must ...
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        votes
        0answers
        36 views

        Degree bounds of polynomial factors

        Given a polynomial over the integers, we can use Dumas theorem to narrow down the degrees of potential factors. Are there any other theorem that give bounds on degrees of factors?
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        vote
        0answers
        75 views

        Primes with given Hamming weight

        If I understand correctly, in the following MO-thread Are There Primes of Every Hamming Weight? two users of the site claim that it has been already proven that, for every sufficiently large $n \in \...
        2
        votes
        1answer
        74 views

        Norms of elements in a quadratic order - can you do it elementarily?

        Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Does there exists an element $\lambda\in \mathcal O$ such that the norm $N(\lambda)$ is not a square? Does there exists an element $\...
        4
        votes
        2answers
        459 views

        Estimate related to the Möbius function

        I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. ...
        3
        votes
        1answer
        95 views

        Quadratic orders embedded in matrices

        Let $\tau$ be a CM point and let $\mathcal O$ be the quadratic order corresponding to the lattice $[\tau,1]$, that is $$\mathcal O =\lbrace \lambda \in \mathbb C: \lambda[\tau,1]\subset[\tau,1]\rbrace....
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        votes
        0answers
        47 views

        How can I find the integral orthogonal group of a given symmetric positive definite form?

        I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix: $$M=\begin{bmatrix} x_1 &...
        5
        votes
        1answer
        250 views

        Cube root of the $j$-invariant

        Let $$\Gamma=\bigg \lbrace \begin{pmatrix} a&b\\c&d\end{pmatrix}\in\Gamma(1):b\equiv c~(\text{mod }3)\text{ or } a\equiv d\equiv 0~(\text{mod }3)\bigg \rbrace.$$ Then $\Gamma$ has exactly ...
        1
        vote
        0answers
        329 views

        Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$

        Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let $\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$ In 2014, in the paper Zhi-Wei Sun, New series for some special values of $L$-functions, ...
        3
        votes
        1answer
        110 views

        Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement

        Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
        4
        votes
        1answer
        191 views

        Size of finite subset of $\mathbb{N}$ such that the sum of reciprocals is a given positive integer

        Let $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum_{j\in S}j^{-1}=k$. ...
        4
        votes
        1answer
        126 views

        Discriminant of a radical extension of a quadratic number field

        Let $K=\mathbb Q(\sqrt 5)$ and $\varepsilon = \frac{3 + \sqrt 5}{2}$ its totally positive fundamental unit (i.e. it generates the subgroup of totally positive units). For any $n \geq 3$, let $L_n = K(\...

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