# 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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12 views

### Reference book between Riemann z function and random matrix

What is a reference book to understand the relation between Riemann z function and random Matrix
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### Invariant factors and commuting matrices over a discrete valuation ring

Let A be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T, Y^T$ be their transpose. Assume that $$p^dA^n\subset Im X+Im Y.$$ Does ...
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### Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions

Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that $p$ does not divide the conductor of $d$, $p$ splits ...
1 vote
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### Radicands of square roots of the 2020s, written in simplest radical form

As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
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### Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?

Is equation $$(x+1)y^2-xz^2=x^3+2x+2$$ solvable in integers? Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...
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### How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets

How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
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### Prove that this equation for natural m and n doesnt have an answer [closed]

$19^(19)=m^3 + n^4$ from $19^(19)$ i mean 19 to the power of 19 i've tried m and n for mod k, k=1,2,...,11 but i haven't reached a solution
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### Power summing function [closed]

f(x,p)=sum(n=1,n<=x,n^p) where p and x are integers. f(x,1)=(x^2+x)/2, and f(x,2)=x(x+1)(2x+1)/6, but what is f(x,p), where p is VERY BIG?
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### Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
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### Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?

In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...
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### Does the interior of Pascal's triangle contain three consecutive integers?

This question defeated Math SE, so I am posting it here. Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$. ...
1 vote
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### Collision among products of binomial coefficients

Let $k, \ell \geq 2$ be positive integers. Do there exist infinitely many positive integers $N$ such that the equation $$\displaystyle \binom{m}{k} \binom{n}{\ell} = N$$ have more than four solutions? ...
1 vote
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### Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$

This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted. Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
309 views

### Cohomology of Shimura varieties before and after completion at some prime

Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
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### Diophantine equations involving recurrence sequences

I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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### Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f,g,h,k,l~(>1)$ are natural numbers? [closed]

Is there any odd natural number $n$ satisfying $n=e^2−f^2=g^2−h^2=k^2−l^2$ and $e^2f^2=g^2h^2+k^2l^2$ where $e,f, g, h, k, l$ are natural numbers greater than 1? The problem is related to some famous ...
1 vote
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### Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves

In this article in Compositio (2011), Munshi proves a mean value result for $$\sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$ where here $f$ is a primitive holomorphic form of level $q$ with ...
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### Linear equation among divisors of a positive integer

Let $a,b,c$ be co-prime positive integers and let $\theta \in (1/4, 1/3)$ be a real number. For each positive integer $k$, does there exist a positive integer $N$ such that the linear diophantine ...
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### Framed bordism and string bordism in 3-dimensions vs topological modular form

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
560 views

### Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?

Is $$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$ where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime? Context: This question came out as a result in ...
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### detecting zeros of Dirichlet $L$-functions

I have been reading papers on zeros of Dirichlet $L$-functions for some of my work and I have a question. Just like the zero detecting method pointed out by Littlewood in his 1924 paper On the zeros ...
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### How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]

I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
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### Rank-24 Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant? [migrated]

A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix: ...
635 views

### Is there a regular pentagon with a rational point on each edge?

This question was asked by Yaakov Baruch in the comments to the question Can a regular icosahedron contain a rational point on each face? It seems that this question deserves special attention.
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### Size and type of a Galois radius: are they correlated?

Say as usual that a non negative integer $r$ is a Galois radius of $n$ of type $(a,b)$ if $(n-r,n+r)=(p^a,q^b)$ for some couple of primes $(p,q)$. As primes tend to get sparser as we progress along ...
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### Central limit theorem for irrational rotations

Let $\alpha$ be an algebraic integer of modulus 1, and $R_\alpha z=\alpha z$. Is $$\lim_{n\to\infty}\frac{\log|\sum_{k=1}^n \Re R_\alpha^k z|}{\log n}=\frac12$$ for all $z\in S^1$? Birkhoff's ergodic ...