Questions tagged [totient-function]

Question 1

Consider the integer equation: $ a^{\varphi(c)} + b^{\mathrm{rad}(a)} = c^{\varphi(b)} $ with the conditions: Integers: $a, b, c > 1$ . Totient: $\varphi(n)$ is the Euler totient function. Radical:...

Question 2

I am considering the following inequality involving Euler’s totient function $\varphi$: For every integer $x>4$, $$ x \le \varphi(x-3) + \varphi(x-2) + \varphi(x-1). $$ $x$ from $5$ to $200{,}000$ ...

Question 3

When thinking about an unrelated problem, I wanted to sum $$\sum_{k\in[\sqrt n,2\sqrt n]}\#\{j\leq k/5:\gcd(k,j)=1\}$$ and I wanted this sum to be $\Theta(n)$. From the wikipedia I can see that if we ...

Question 4

Erdős problem 409 ("How many iterations of $n↦\phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?...

Question 5

Consider the nonlinear recurrence $$x_n=1+\frac{\phi(x_{n-1})+\phi(x_{n-2})}2,\;\;\;x_0,x_1\gt2$$ where $\phi$ is Euler's totient function. The only possible fixed points $C$ of the sequence satisfy $$...

Question 6

Consider the following three theorems: If $m,n$ are relatively prime, then $\varphi(mn) = \varphi(m)\varphi(n)$ (Where $\varphi$ is the totient function, the Euler-phi function) If $f: A \rightarrow ...

Question 7

The system is given as: $$ \begin{matrix} x \equiv _{107} 52!53! \\ x \equiv _{16} 11^{9^{13}} \end{matrix}$$ $gcd(16, 107) = 1$ so Chinese remainder theorem can be applied. However, my issue are the ...

Question 8

I am trying to use polynomial roots to approximate this ratio $R(p)$: $$R(p)=\Re\left(\frac{\underset{k\to \infty}{\text{lim}}\left(\left(H_k^{(s)}\right)^{1/p}+\left(\frac{k^{1-s}}{s-1}\right)^{1/p}\...

Question 9

I know of different types of solutions to relations between $\varphi(n)$ and $n$, e.g. the classical question to find all positive integers $n$ for which $\phi(n)$ divides $n$. But the following ...

Question 10

Let $$2^{\ell-1}<p_1<p_2<\dots<p_t<2^\ell$$ $$2^{\ell'-1}<q_1<q_2<\dots<q_m<2^{\ell'}$$ be primes on the condition $$\phi(p_i)=2q_1^{a_1}\dots q_m^{a_m}$$ ($\phi$ is ...

Question 11

I was watching a Udemy course on Number Theory and the instructor presented the following argument which I cannot follow. For example, how are there p - 1 numbers in the last row? Is there a typo ...

Question 12

In a programming contest held yesterday, there was this one question: A number $N$ is called good if its prime factorization consists of all primes with atleast $2$ digits. For example $1111 = 11 \...

Question 13

While studying the following theorem: If $m_1$ and $m_2$ are two positive, relatively prime integers, then $$ \varphi(m_1) \varphi(m_2) = \varphi(m_1 m_2). $$ I decided to explore the values of $\...

Question 14

I want to show that $$ \sum_{\substack{n\leq x\\(n,k)=1}}\frac{1}{n}=\frac{\phi(k)}{k}\log(x)+O(1) $$ I think that the proportion of number less than or equal to $x$ that are coprime to $k$ is $\phi(k)...

Question 15

I studied the Euler totient function $\phi(k)$ and found on Wikipedia that $$ \sum_{k=1}^n \phi(k) = \frac{3}{\pi^2} n^2 + O(n \log n). $$ Therefore, $$ \lim_{n \to \infty} \frac{\zeta(2)}{n^2} \sum_{...

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Questions tagged [totient-function]

Solutions of $ a^{\varphi(c)} + b^{\mathrm{rad}(a)} = c^{\varphi(b)} $ [closed]

Inequality involving Euler's totient function on three consecutive integers

Restricted Euler function

How to extend Erdős problem $409$ to $\mathbb{N}^3$?

A sequence based on Euler's totient function that always converges to prime numbers

Connections between similar looking theorems

How to solve the system involving factorials? [duplicate]

Using the analytic continuation of the Riemann zeta function to approximate some polynomial roots.

Solutions to a relation between $\varphi(n)$ and $n$

Common element with order constraints

Proof of formula for Euler's Totient Function for a prime power [closed]

Counting numbers which don't contain single digit primes in their prime factorization [duplicate]

Is $\gcd(\varphi(2^m),\varphi(3^m),\varphi(4^m),\varphi(5^m),\dots )=2$? [closed]

Prove that $\sum\limits_{\substack{n\leq x\\(n,k)=1}}\frac{1}{n}=\frac{\phi(k)}{k}\log(x)+O(1)$ [duplicate]

How to prove the general limit of weighted sums involving Euler’s totient function?

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