MHB Number of natural numbers that have primitive roots

AI Thread Summary
The discussion focuses on calculating the number of natural numbers between 2 and n that have primitive roots, which occurs when m is cyclic. A number m has a primitive root if it is of the form 1, 2, 4, p^k, or 2·p^k, where p is a prime. To find the count of such numbers, one must determine the number of primes between 2 and n^(1/k). While the density of primes can provide approximations, it cannot yield exact counts, leading to the suggestion of expressing results in terms of the number of primes within a specific range. The limit of the ratio of these counts to n can be approached using prime density without needing exact counts.
mathmari
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Hey! :giggle:

How can we calculate the number of natural numbers between $2$ and $n$ that have primitive roots?

Let $m$ be a positive integer.
Then $g$ is a primitive root modulo $m$, with $(g,m)=1$, if the modulo of $g\in (Z/m)^{\star}$ is a generator of the group.

We have that $g$ is a primitive root modulo $m$ if it is a generator of a group, i.e. $m$ has a primitive root if $\mathbb{Z}_m$ is cyclic, right?

$\mathbb{Z}_m$ is cyclic if $m=1,2,4$ or $m=p^k$ or $m=2\cdot p^k$ for $p$ prime.

That means that the number of natural numbers that have a primitive root is $\#\{1,2,4,p^k, 2\cdot p^k\}$ for $p$ prime.

So we have to calculate the number of primes between $2$ and $n^{\frac{1}{k}}$ to calculate then the number of elements of the form $p^k$ and $2\cdot p^k$.

Have I understood that correctly? :unsure:
 
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Looks about right. :unsure:
 
Klaas van Aarsen said:
Looks about right. :unsure:

To calculate the number of these primes do we use the density of primes? :unsure:
 
mathmari said:
To calculate the number of these primes do we use the density of primes?
We can only approximate the density of primes.
So we cannot use it to find an actual number.
Assuming that we want a 'hard' number, I think we should express it in terms like 'the number of primes between $2$ and $n$'. :unsure:
 
Klaas van Aarsen said:
We can only approximate the density of primes.
So we cannot use it to find an actual number.
Assuming that we want a 'hard' number, I think we should express it in terms like 'the number of primes between $2$ and $n$'. :unsure:

Actually I want to calculate the limit $\displaystyle{\lim_{n\rightarrow \infty}\frac{a_n}{n}}$ where $a_n$ is the above number. So do we need the actual number to calculate this limit? :unsure:
 
mathmari said:
Actually I want to calculate the limit $\displaystyle{\lim_{n\rightarrow \infty}\frac{a_n}{n}}$ where $a_n$ is the above number. So do we need the actual number to calculate this limit?
No. I think we can use the density of primes to calculate that limit. :unsure:
 
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