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 role, if any, do Archimedean valuations play in adic spaces?

I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...? Is there a weird ...
Andrew NC's user avatar
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7 votes
1 answer
331 views

Conjectured combinatorial non-equality

Let $n,k,\ell$ be integers for which $0\leq k<\ell \leq n-6$. For a fixed $n$, think of $k,\ell$ as being allowed to vary. I believe the values $$(n-k-5)(k+1)(k+2)\binom n{k+3}~~~\text{and}~~~(n-...
John McVey's user avatar
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6 votes
0 answers
321 views

Embeddings of number fields into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$

When studying arithmetic Galois representations for a number field $F$ one often fixes at the outset an embedding of its algebraic closure $\bar{F}$ into $\mathbb{C}$ or $\bar{\mathbb{Q}}_l$ and ...
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3 votes
0 answers
123 views

Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$

We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...
Jesse Railo's user avatar
1 vote
1 answer
142 views

About a Dirichlet series [closed]

I would like to know if the following assertion is true: Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
teller's user avatar
  • 337
8 votes
1 answer
4k views

Number of integer solutions of a linear equation under constraints

How many positive integer solutions of $$\sum_{i=1}^{k}x_i = N$$ for some positive integer $N$ given the constraints $n_i\leq x_i\leq m_i$ for $i=1,\ldots,k$, where $n_i$ and $m_i$ are positive ...
Satya Prakash's user avatar
7 votes
1 answer
308 views

Rational perfect power values of $y(y+1)$

This is hard, so I am looking for partial results and how hard it is. Let $n>4$. Is it true that the hyperelliptic curve $x^n=y(y+1)$ doesn't have rational point with $x \ne 0$? If necessarily ...
joro's user avatar
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0 votes
0 answers
159 views

Counting divisors of primes plus or minus 2

I was naively exploring the (usually) composites $\pm 2$ from a prime $p$, wondering if there might be some asymmetry, and made this histogram of the difference $\Delta$ in the number of divisors of $...
Joseph O'Rourke's user avatar
4 votes
0 answers
146 views

Generalization of a determinant with Lucas numbers and totient functions

Let $\gcd(a,b)$ denote the greatest common divisor function. H. J. S. Smith proved that $$\det\left[\gcd(i,j)\right]_{i,j=1}^n=\prod_{k=1}^n\varphi(k),$$ where $\varphi(k)$ denotes Euler's totient ...
T. Amdeberhan's user avatar
5 votes
1 answer
201 views

Collapsed partitions and generating functions

Given $n\in\Bbb{N}$, the number of (unrestricted) integer partitions of $n$ are given by $$\sum_{n\geq0}p(n)x^n=\prod_{j\geq1}\frac1{1-x^j}.$$ Define the collapsed partitions of $n$ to be the ...
T. Amdeberhan's user avatar
1 vote
2 answers
346 views

Hyperelliptic curves imply FLT-like results

Probably this is known, but doesn't show in searches. If a certain hyperelliptic curve has only trivial rational points, FLT-like curve also has only trivial rationals points for fixed $n$. Working ...
joro's user avatar
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14 votes
1 answer
862 views

BSD conjecture for rank 1 elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that $$\text{ord}_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$ Thanks to the work of Gross-Zagier and ...
baobab's user avatar
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31 votes
2 answers
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Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture. Goldbach's conjecture asserts that every ...
Tanner Swett's user avatar
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4 votes
1 answer
939 views

Arithmetic properties of a sum related to the first Hardy-Littlewood conjecture

The starting point of this post is an earlier question, where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at $s=1$ of a certain Dirichlet series, $$\Lambda(m)=\...
Mats Granvik's user avatar
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-1 votes
2 answers
367 views

Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?

Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research. There do not exist $a,b$ and $c$ such that$$ (2a-1)(2^{(b+c)}-3^c )=2^b-1 $$where $a>0,b&...
MathAllTheTime's user avatar
5 votes
0 answers
233 views

No rational points on $x^n+a=y^2$ for all $n>4$"?

Is there rational (or better integer) $a$ such that for all $n>4$,$x^n+a=y^2$ has no rational points?
joro's user avatar
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1 vote
0 answers
142 views

On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$

For $n>4$ and coprime integers $x,y,z$ consider the diophantine equations: $$x^4+16z^n=y^2 \qquad (1)$$ and $$x^4+z^n=y^2 \qquad (2)$$. (2) is special case of Fermat Catalan and is solved. For ...
joro's user avatar
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3 votes
1 answer
365 views

Primality test for specific class of $N=k \cdot b^n-1$

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ . Can you provide a proof or a counterexample to the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(...
Pedja's user avatar
  • 2,673
1 vote
0 answers
104 views

1-concatenable primes

If we choose some prime, say $11$, we can concatenate one digit to the left and one to the right to obtain another prime, for example, $2113$, we can do the same with $2113$ and obtain $121139$, a ...
Right's user avatar
  • 225
0 votes
1 answer
277 views

Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]

Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$ \ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
Nourddine Snanou's user avatar
6 votes
0 answers
266 views

Examples in which probabilistic heuristic reasoning fails

There are examples of conjectures in which one can use probabilistic heuristic reasoning to show that they are very likely to be true. For instance, Freeman Dyson used probabilistic heuristic ...
Craig Feinstein's user avatar
20 votes
1 answer
882 views

Double Counting: Motivic Edition

One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
Gjergji Zaimi's user avatar
50 votes
5 answers
3k views

Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
Sarah's user avatar
  • 482
5 votes
1 answer
297 views

How to show that a hypersurface is a diagonal intersected with hyperplanes?

Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(...
Johnny T.'s user avatar
  • 3,547
6 votes
2 answers
385 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
Jesse Elliott's user avatar
8 votes
0 answers
253 views

Order of magnitude of extremely abundant numbers and RH

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
Wolfgang's user avatar
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4 votes
1 answer
200 views

Local L-function $L(s,\pi_p\times \chi_p)=1$

Let $\pi_p$ be a ramified representation of $GL(n,\mathbb{Q}_p)$. Let $\chi_p$ be a ramified representation of $GL(1,\mathbb{Q}_p)$. Is it generally known that $L(s,\pi_p\times \chi_p)=1$ if $\...
7-adic's user avatar
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13 votes
0 answers
356 views

Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients

Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
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17 votes
0 answers
956 views

Why arithmetic Langlands?

In trying to understand the import of Akshay Venkatesh' most recent work I found myself wondering anew about that old gnawing mystery: why Langlands? Why should arithmetic of polynomial equations over ...
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1 vote
0 answers
107 views

Counting "simultaneous squares' over the Gaussian integers

Let $n$ be a square-free integer. Then for a given integer $m$, $m$ is a square modulo $n$ if and only if the sum $$\displaystyle \sum_{d | n} \left(\frac{m}{d}\right) > 0.$$ In fact one can ...
Stanley Yao Xiao's user avatar
3 votes
1 answer
268 views

How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$ First question What can we say in general about the factor $j(\...
Shimrod's user avatar
  • 2,335
48 votes
6 answers
5k views

Are there examples of conjectures supported by heuristic arguments that have been finally disproved?

The idea for this comes from the twin prime conjecture, where the heuristic evidence seems just so overwhelming, especially in the light of Zhang's famous result from 2014 about Bounded gaps between ...
2 votes
0 answers
154 views

Categorical representations of absolute Galois groups

I am looking for interesting examples of categorical representations of absolute Galois groups of arithmetic fields. Pointers to the literature would be appreciated.
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3 votes
0 answers
155 views

The name for injective map $f:\mathbb{N}\rightarrow\mathbb{N}$ with $f(n)\geq n$ property

What is the name for map $f:\mathbb{N}\rightarrow\mathbb{N}$ (from natural numbers into natural numbers) with the following propeties: 1) $f$ is injective 2) $f(n)\geq n$ for every $n$?
Lyudmyla Polyakova's user avatar
3 votes
0 answers
160 views

Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
Turbo's user avatar
  • 13.7k
3 votes
1 answer
408 views

What is the shortest length of an Egyptian fraction expansion for a given $p/q$?

An Egyptian fraction expansion is a sum of reciprocals of integers, for example: $$\frac{4}{17} = \frac{1}{5} + \frac{1}{29} + \frac{1}{1233} + \frac{1}{3039345}$$ Every positive rational number $p/...
Kim's user avatar
  • 4,034
0 votes
1 answer
157 views

Parity and number of squares taken by polynomials in a range?

I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
Turbo's user avatar
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34 votes
1 answer
2k views

On a quantum Riemann Hypothesis

Here is a revised version: On a revised quantum Riemann hypothesis. Robin's theorem (1984) states that $$ \sigma(n) < e^\gamma n \log \log n$$ for all $n > 5040$ if and only if the Riemann ...
Sebastien Palcoux's user avatar
3 votes
0 answers
191 views

Automorphy of families of motives

I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families. Suppose we have an algebraic family of varieties over a number field, and ...
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3 votes
0 answers
232 views

Visualization of hidden structures in numbers

[Please allow me a note: The way desribed below allows to depict functions $f:X^2 \rightarrow Y$ completely in two dimensions (without hiding or omitting any information). This allows for depicting ...
Hans-Peter Stricker's user avatar
9 votes
1 answer
1k views

Are there infinitely many primes of this form?

The semiprime $87 = 3*29$ has a curious property: it's the fact that both $87^2 + 29^2 + 3^2 = 8419$ and $87^2 - 29^2 - 3^2 = 6719$ are prime numbers. This intrigued me and led me to wonder if ...
E.F's user avatar
  • 91
4 votes
2 answers
358 views

A specific Diophantine equation restricted to prime values of variables.

Consider the following Diophantine equation $$x^2+x+1=(a^2+a+1)(b^2+b+1)(c^2+c+1).$$ Assume also that $x,a,b,c,a^2+a+1,b^2+b+1, c^2+c+1$ are all primes. We'll call such a quadruplet $(x,a,b,c)$ a ...
JoshuaZ's user avatar
  • 6,090
10 votes
0 answers
362 views

Recognizing the Galois group from the field discriminant

Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
Joachim König's user avatar
7 votes
1 answer
214 views

Is anything known about this class of series involving the divisor function?

I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it! Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
Alex Saad's user avatar
  • 663
24 votes
1 answer
585 views

Has the $E_8$-based generating function for squares numbers been proven?

In his 2004 paper Conformal Field Theory and Torsion Elements of the Bloch Group, Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $...
Theo Johnson-Freyd's user avatar
0 votes
1 answer
261 views

Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product \begin{equation*} \prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
Junhyeong Kim's user avatar
0 votes
0 answers
88 views

Infinite difference length of integer subsets

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition: We say $A$ has infinite difference length, if (a) For every integer $n$ there exist a ...
M.H.Hooshmand's user avatar
4 votes
1 answer
316 views

Higher roots modulo prime complexity best algorithm

Given integers $a,\ell$ and prime $p$ we need to find the roots of the algebraic equation $x^\ell\equiv a\bmod p$. We know there are at most $\ell$ such $x$. What is the best method to find all such ...
Amal Duriseti's user avatar
2 votes
0 answers
254 views

The Hilbert Symbol and real algebraic geometry

Let $(a,b)_K$ be the quadratic Hilbert symbol in a local field $K$. Let $a$ be a rational number. By a consequence of the quadratic reciprocity law we have: $$\prod_{p} (a,-1)_{\mathbb{Q}_p}=\mathrm{...
camilo's user avatar
  • 527
2 votes
1 answer
206 views

Reference request: the dual Coleman family

Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an ...
GRH's user avatar
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