# 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.

262 views

### Proof regarding Robin's inequality (RI).

Let $\sigma$ be the divisor sum function, $\gamma$ the Euler-Mascheroni constant and $n>5040$. Robin showed that if the inequality$$\displaystyle \sigma(n)<e^{\gamma}n\log\log n$$ ever fails, it ...
34 views

### $\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ solve for positive integers. [duplicate]

Solve for positive integers $x,y,z$ $$\frac{4}{13}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ I tried to solve it by some generel work but it didn't help.
117 views

### Can sieve method prove ternary (three) prime Goldbach conjecture?

Can sieve method prove ternary (three) prime Goldbach conjecture (Vinogradov Theorem) ? I had done some research, I could not find any articles on this. Can anyone provide some help on this ? I ...
842 views

### Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some prime ...
102 views

### Find two positive integers $x$ and $y$ such …

Find two positive integers $x$ and $y$ such $$\sqrt{69+20\sqrt{11}}=\sqrt{x}+\sqrt{y}$$ I have worked intensively with this task but I really can't find a solution to this problem. I hope that I ...
183 views

### My (divergent) summation of the zetas with sets of cofactors give systematically errors of simple integer differences. What am I missing?

This is a "fiddling" in a small project of mine with which I'm concerned from time to time for three years now. I try to focus on the core of the problem, please ask if more context is needed. ...
37 views

170 views

### The frequency occurrence of primes in the unique prime factorization of natural numbers.

I am curious about the frequency of occurance of prime numbers in natural numbers. For example starting with the first non-prime 4 = 2^2, then 6 = 2x3, 8 = $2^3$, 9 = $3^2$ etc. Now of course a prime ...
159 views

### Are fractions with prime numerator and denominator dense?

So the question is in the title. And I mean dense in positive real numbers ofcourse. Somehow I cannot grasp if this is very trivial or not. The prime numbers aren't that dense, but are there enough of ...
148 views

### Proof of $\sum_{n=1,3,5,\ldots}^{\infty}\frac{1}{n^4}=\frac{\pi^4}{96}$

I came across with the infinite series $$\sum_{n=1,3,5,\ldots}^{\infty} \frac{1}{n^4}= \frac{\pi^4}{96}$$ when calculating a problem about an infinite deep square well in quantum mechanics. ...
46 views

52 views

### How to prove the $q$-series identity?

How to prove the following identity: $$\sum_{n\ge0}\frac{2q^{n^{2}+n}}{(q)_{n}^{2}(1+q^{n})}=\sum_{n\ge0}\frac{q^{n^{2}+n}}{(q)_{n}^{2}(1-q^{2n+2})}$$
43 views

### Is there a simpler way to rewrite this binomial chain?

Consider some binomial chain that looks like this: $$\binom{N}{k_1}\binom{N-k_1}{k_2}\binom{N-k_1-k_2}{k_3}\binom{N-k_1-k_2-k_3}{k_4} \cdots \binom{N-k_1-k_2-\cdots -k_{t-1}}{k_t}$$ Where all ...
73 views

### Is there any simple algorithm which can tell if a string is a repeat of its substring?

Is there any simple algorithm which can tell if a string is a repeat of its substring? For example, $1212121212$ is a repeat of $12$, $135746135746$ is a repeat of $135746$.
36 views

### Square root in $\mathbb{Z}_p{}^*$

Given a prime $q$, and another prime $p$ = 20q + 1, I am able to find generators in $\mathbb{Z}_p$. Does $-1$ have a square root in $\mathbb{Z}_p{}^*$? Thanks!
207 views

### Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
86 views

### Decide whether the $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ is rational

Working needs to be shown $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ My guess is to multiply by $\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}$ then we have a rational number but is it enough to prove the ...
2k views

157 views

### How find this limits $\lim_{n\to\infty}(\frac{2}{1^4}+1)(\frac{2}{2^4}+1)(\frac{2}{3^4}+1)\cdots(\frac{2}{n^4}+1)$

How to Find this limit $$\lim_{n\to\infty}\left(\dfrac{2}{1^4}+1\right)\left(\dfrac{2}{2^4}+1\right)\left(\dfrac{2}{3^4}+1\right)\cdots\left(\dfrac{2}{n^4}+1\right)$$ see 1 I remeber ...
118 views

### $x^{16}-16 \equiv 0 \mod p$ has a solution for each prime

I have to prove that $x^{16}-16\equiv 0 \mod p$ has a solution for every prime $p$. I already know (from a previous work) that $x^8-16\equiv 0 \mod p$ has a solution for every prime. In my opinion, I ...
102 views

55 views

### Möbius function matrix

In http://en.wikipedia.org/wiki/M%C3%B6bius_function#Matrix_inverse, it is stated, that the values of the Möbius function appear in the inverse matrix of some lower triangular matrix. Does anyone know ...