I’ve tried proving that $\varphi(mn) = \varphi(m)\varphi(n)$ (if $gcd(mn)=1$). The proof I try to setup doesn’t look like the proof I find in textbooks, where am I going wrong? Proof: We try to show that the number of relative primes to $mn$ in $\{1, 2, \ldots , mn-1\}$ is equal to the number of relative […]

In the work on another question in MSE I have a formula $f(n)$ whose pattern depending on $n \in \Bbb N$ I want decode into an algebraical formula (see a short rationale of $f(n) at the end). Beginning with the most simple $n$ – namely that which consist of pairs of primes $p,q$ with $n=pq$ […]

I was skimming again through Dummit & Foote’s Abstract Algebra and I came across this exercise: Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n)=N$, where $\varphi$ denotes Euler $\varphi$-function. Conclude in particular that $\varphi(n)$ tends to infinity as $n$ tends to infinity. I don’t doubt that […]

For calculating $$a \uparrow \uparrow n\ (mod\ m)$$ the chain $$m , \phi(m) , \phi(\phi(m)) , \phi(\phi(\phi(m))) , … $$ is useful. As $\phi^n(m)=1$ for some n, the above modulo calculation gets stationary at some point. So, it would be useful, if the length of the phi-chains could be suitable bounded. Which lower and upper […]

If we divide Euler’s totient function $\omega(n)$ by $n$, we obtain a fraction. If we multiply this fraction by any natural number $m$ which gives us another natural number $p$, is it true that $p$ is the number of natural numbers in $[1,m]$ that are coprime to $n$?

Ok, I really need some help understanding this because either my brain isn’t working at the moment or I’m breaking math and I have a striking suspicion that one of those is more likely. Anyways, here’s my process. WolframAlpha tells me there are 12 integer solutions, they are all $^+_-n$ so let’s say there are […]

How do I find the equation that will give me the result for $\phi (n)=10$ and find any possible values of $n$ that works?

I’m trying to estimate asymptotically the following sums: $$ S_1(m, n) = \sum_{1\leq i \leq n, (m,i)=1}{\frac{1}{i}} $$ $$ S_2(n) = \sum_{i=1}^n{i\phi(i)}, $$ where $(m,i) = GCD(m,i)$, and $\phi(i)$ is the Euler totient function of $i$. I would be grateful for any hints.

As my most recent question still does not have any answers and it appears to be a difficult problem, I propose the following problem (that seems easier), but which I still could not manage to solve: Is it true that for every $k \in \mathbb{N}$ there exists distinct natural numbers $x_1, \cdots, x_k$ such that […]

I’ve seen the graph for $\varphi(n)$ vs $n$ on Wolfram, and it seems like the $\varphi(n)$ values for primes follow a constant slope, but is there a proof that states that the totient values between two consecutive primes are always smaller than the totient values of those two primes? EDIT: I got a satisfactory answer […]

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