# 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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### What are the consequences if the Birch–Swinnerton-Dyer conjecture is assumed to be false

The Birch–Swinnerton-Dyer (BSD) conjecture (https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture) describes the set of rational solutions to equations defining an elliptic curve. This ...
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### $\exists k$ s.t. $k^m\le 1^m+2^m+…+(k-1)^m <2\cdot k^m$?

Can it be shown that,for all $m\in\mathbb{Z}_+$ there exists at least one $k$ with respect to $m$ such that $$1\le \frac{\sum_{i=1}^{k-1}i^m}{k^m}<2$$ Example: let $m=1$ then $k=\{3,4\}$ This ...
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### Estimating certain short Kloosterman sums

Recall that for the classical Kloosterman sum $$K(a,b,p^t):= \sum_{x \in (\mathbb{Z}/ p^t \mathbb{Z})^* } \psi \left(\frac{ax+bx^{-1}}{p^t} \right),$$ where $\psi(x)=e^{2\pi ix}$, $a,b,t$ are natural ...
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### Geometry of algebraic curve determined by point counts over all number fields?

Let $C$ be a smooth irreducible projective curve of genus $g>1$ over a number field $K$. The Mordell conjecture (first proved by Faltings) says that for any finite field extension $L/K$ (say, ...
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### Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$

For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
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### Generating all graphs of order 4 with the help of Collatz

Given a set of positive integers, its common divisor graph ( CD-graph) is the graph whose vertices are the integers, two of which are joined by an edge if (and only if) they have a common divisor ...
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### Numerical estimates for a function relating to twin primes :

Consider the following function : $$F(s)= \sum_{\text{p,\ p+2 are primes}} \left({\frac{1}{p^s}}+{\frac{1}{(p+2)^s}}\right).$$ Brun's theorem tells us that $F(1)$ is finite. We are looking for ...
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### Does this equation have more than one integer solution?

Consider the following diophantine equation $$n = (3^x - 2^x)/(2^y - 3^x),$$ where $x$ and $y$ are positive integers and $2^y > 3^x$. Does $n$ have any other integer solutions besides the case ...
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### The abc-conjecture over the positive rationals and Levy-Schoenberg kernels?

I am continuing the "abc-adventure" and have a specific question, which needs some explanation: In this paper by Gangolli, the term "Levy-Schoenberg" kernel is defined (Definition 2.3). Consider the ...
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### On Kellner's result and the Erdos-Moser equation

Let $m, k$ be positive integers. Consider the Erdos-Moser expression $S_{k}(m) = 1^k + 2^k + ... + (m-1)^k$. By a result of Kellner, we know that if $m | S_{k}(m)$, then $m|B_k$, where $B_r$ denotes ...
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### Primes mod 4 and integer polynomials

I have asked these questions as comments here (these are related to the question there). The questions are: Let $S$ be one of the following sets of primes: All primes of the form $4k+1$ ; All primes ...
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### Schur polynomials with zeros in an infinite geometric progression

Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$. Question: Is ...
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### Distribution of signs of automorphic forms

Let's say we have an automorphic form $f$ on $GL(2)$ that is self-dual. In particular, the associated L-function $L(s,f)$ satisfies a functional equation with sign $\varepsilon_F = \pm 1$. Is it ...
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### Relations between spectral parameters of automorphic representations

Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated ...
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### Bijection map between equation and integer point [closed]

Suppose that we have a equation E define by $n!=m^2(m^2-2)+1$ and we want to solve it in $\mathbb{N}$ , and transforme it by the bijection map $\phi$ : $(n,m )$ $\rightarrow$ $(n,2m)$ to ...
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### For what automorphic representations is Ramanujan-Petersson known?

I had in mind that Ramanujan-Petersson conjecture was essentially unknown in the case of number fields. I however recently heard that If an automorphic representation on $GL(2)$ is ramified at a ...
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### Which Hilbert's 10th polynomials are known to have solutions?

The Diophantine equation $$x^3 + y^3 + z^3 = 42$$ was recently solved by Booker and Sutherland: Sum of three cubes for 42 finally solved. Is there a clean partition of the form of those polynomial ...
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### Asking for reference request to study the proof of a result which is used in atleast 4 papers to prove existance of irrational odd zeta values

I am studying a research paper of T. Rivoal and Wadim Zudilin , "a note on odd zeta values " and I am unable to think how a result implies the theorem to be proved. So, I began to read other paper of ...
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### Limit of a ratio of harmonic numbers?

Is there any way to find the following limit $$R(n,m)=\lim_{N\to\infty}\frac{H_{nN,m}}{H_{N,m}}$$ which involves harmonic numbers (generalized if $m\neq 1$) $$H_{N,m}=\sum_{k=1}^N k^{-m}\qquad ?$$ ...
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### Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics

Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-...
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### Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
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### Nature of Fourier coefficient of a modular form after applying a certain map (trace operator)

Asking this here because of no response at (MathStackExchange). Let $N|M$, and consider the trace operator $Tr^M_N$ defined on $M_k(\Gamma_0(M))$ - vector space of modular forms of weight $k$ for ...
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### Brocard's conjecture for Ramanujan primes

Brocard's conjecture is a conjecture about the expected number of prime numbers $p_k$ between the squares of two consecutive prime numbers, I add as reference the Wikipedia Brocard's conjecture or ....
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### Neukirch's theorem on absolute Galois groups in English [duplicate]

Is there a paper or book available in English that proves the result of Neukirch on absolute Galois groups of number fields? I'm having a hell of a time with the German originals.
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### Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?

Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
Let $a$ and $b$ be two relatively prime positive integers. Denote by $S$ the set of all positive primes of the form $a+bn$ (where $n$ is an integer, possibly zero or negative). Let $S'$ be any set of ...