# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2$ in the ...
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### A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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### Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant ...
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### Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
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### Representing of natural number

Let $n$ be a given natural number with $n>1$, please prove that each natural $x>=2n$ can be represented as $$x=a_1^n+a_2^n+\cdots+a_n^n-(b_1^n+\cdots+b_{n-1}^n)$$ where ...
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### To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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### 27 lines on a smooth cubic surface

It is known that every smooth cubic surface with coefficients in $\mathbb{Q}$ has $27$ lines defined over a number field extension of $\mathbb{Q}$ of degree at most $51840$ as the group ...
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### There exist infinitely many positive integers $k>r!$, such that for any $j$ with $r!<j<k$ we have $\prod_{i=0}^{r-1}(j-i)\nmid\prod_{i=0}^{r-1}(k-i)$

Question: Let $r$ be a postive integer. Show that there exist infinitely many positive integers $k$ satisfying $k>r!$,such that for any positive integer $j$ satisfying $r!<j<k$ we have ...
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### Can we use $n\log n$ instead of $n$-th prime?

Denote $\pi(x)$ be the number of primes $\leq x,$ $p(n)$ be the $n$-th prime number. We have $\pi(p(n))=n.$ It's well known that $$\pi(x)\sim \frac{x}{\log x} \\p(n)\sim n\log n.$$ Is it always ...
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### How to list the prime factorised natural numbers?

Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
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### Coprime multiplicative orders modulo infinitely many primes

Is it true that there are infinitely many primes $p$ such that the multiplicative orders of $2$ and $3$ are coprime $\pmod{p}$? By this I mean their order in $(\mathbb{Z}/p\mathbb{Z})^*$. If the ...
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### About solutions to $x^2+y^2+z^2=8n+3,n\in \mathbb N$

As we know that $x^2+y^2+z^2=8n+3,(n\in \mathbb N)\tag{1}$ has integer solutions $x,y,z\in \mathbb N.$ If $k\in \mathbb N$ has at least one prime factor which is $\equiv 3 \mod 4,$ then we call $k$ ...
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I don't know much about number theory, it seems this summation might involve some facts from number theory. Could you give me some idea of doing it? Thank you very much. The summation is $$... 0answers 434 views ### Sum of reciprocal of primes in arithmetic progression In http://www.math.dartmouth.edu/~carlp/Lehmer0.5.pdf on page 6 (top) the author states that:$$ \sum_{p \le x, \ p \equiv 1 \bmod l} \frac{1}{p} = \frac{\log \log x}{\phi(l)} + O \left ( \frac{\log ...
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A Woodal number is an integer of the form $n 2^{n}-1$. A Woodal prime is an integer that is both a prime and a Woodal number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ( ...
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### Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
### About the isomorphism $\operatorname{Br}(\mathbb{Q}_p)\cong \mathbb{Q}/\mathbb{Z}$
I've been reading the section about Brauer groups in Introduction to Modern Number Theory, and I couldn't quite understand how this isomorphism is defined. We start with a central simple algebra $A$, ...