MHB Number of natural numbers that have primitive roots

mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
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:
 
Mathematics news on Phys.org
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:
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top