## Why shouldn’t this prove the Prime Number Theorem? [on hold]

Denote by $$\mu$$ the Mobius function. It is known that for every integer $$k>1$$, the number $$\sum_{n=1}^{\infty} \frac{\mu(n)}{n^k}$$ can be interpreted as the probability that a randomly chosen integer is $$k$$-free.

Letting $$k\rightarrow 1^+$$, why shouldn’t this entail the Prime Number Theorem in the form

$$\sum_{n=1}^{\infty} \frac{\mu(n)}{n}=0,$$

since the probability that an integer is “$$1$$-free” is zero ?

## A nilpotent primitive group is a cyclic group of prime order

There is an exercise in Peter Cameron’s Permutation Groups that a nilpotent primitive group is cyclic of prime order. However I can only prove the finite case, by writing the group as a direct product of its Sylow subgroups. How can I use the nilpotency if the group is infinite?

## on generating prime numbers

Are there yet any known reliable methods of generating the prime numbers less then some arbitrary positive integer which do not involve the use of Sieve of Eratosthenes or divisibility testing?

## Positive integers sum of two positive integers relatively prime to $n$

Let $$n$$ be a positive integer relatively prime to $$6$$. Let $$S_n$$ be the set of positive integers which are relatively prime to $$n$$. Consider the sumset $$2S_n:=\{a+b, a,b \in S_n\}$$. For which $$n$$ (if such a $$n$$ even exists) do we have $$2S_n \neq \mathbf{N}$$?

## The prime spectrume of integral-valued polynomial ring

Let $$D$$ be an integral domain with quotiont field $$K$$ and let $$Int (D)$$be the set of all integral-valued polynomials on $$D$$, that is, $$Int (D):=\{f \in K[x]\mid f (D) \subseteq D\}$$. The structure of prime spectrum of $$Int (D)$$ is well-known for a Noetherian local one dimensional domain. Is there any characterization for the prime spectrum of $$Int (D)$$ without Noetherian local one dimensional assumptions? Thanks for any help.

## A strange sequence of numbers that reversed are prime

$$1, 17, 751, 967, 32953,…$$ is a sequence that I invented.

The first term of the sequence divides $$10^1+12345679$$ The second term divides $$10^{17}+12345679$$.

The n-th term of the sequence a(n) divides $$10^{a(n)}+12345679$$.

Apart the first term, if you reverse the other four terms of the sequence, you have a prime. I don’t know if other terms of the sequence could be calculated. $$12345679=37\cdot 333667$$. Is this sequence present in Oeis and has some importance? Do you believe that if other terms exist, reversed are primes?

## Safari won’t play Amazon Prime Video on my iMac?

If I try to play videos on my iMac using Safari it won’t play, but it plays fine using Chrome. I’m using macOS High Sierra and I’m running Safari version 12.1.

When I try using Safari I get a black screen with a message as shown below.

Can anyone tell me what I need to do to get Safari to play videos?

I don’t think I had Silverlight installed but I don’t know how to tell. I went ahead and installed version 5.1 of Silverlight but again I can’t tell if its installed. When I go to Safari preferences and click on security I get the pane shown below.

## How do I change how much spacing there’s between icons with Nova Launcher Prime?

I don’t find an option to reduce the amount of spacing there’s between icons with the Nova Launcher Prime.

In this screenshot you can see a lot spacing between icons that I don’t want: https://photos.app.goo.gl/swBC25Fk49YHqUEL9

## Is the language { | p and n are natural numbers and there’s no prime number in [p,p+n]} belongs to NP class?

I was wondering if the following language belongs to NP class and if its complimentary belongs to NP class:

\begin{align} C=\left\{\langle p,n\rangle\mid\right.&\ \left. p \text{ and n are natural numbers}\right.\ &\left.\text{ and there’s no prime number in the range}\left[p,p+n\right]\right\} \end{align}

I am not sure, but here’s what I think: for each word $$\langle p,n\rangle \in C$$ we know that the word belongs to C because there exists a primal certificate – an nontrivial divisor to any of the numbers between $$[p,p+n]$$, though I am not really sure it is in NP.

regarding the complement: I think it is in NP because the compliment compositeness can be decided by guessing a factor nondeterministically. But again I am not so sure about it and I don’t know how to correctly prove and show it.

Would really appreciate your input on that as I am quite unsure and also checked textbooks and internet (and this site) about it.

## Units, Prime Elements and Irreducible elements for polynomials

Are there any general steps or things to look out for when identifying the unit elements, irreducible elements and prime elements in ring of polynomials eg Z_4[x], C[x], R[x], Q[x].

It seems a little tricky.

Thank you.