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Question 1

2votes

1answer

95views

Inequality involving Euler's totient function on three consecutive integers

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 $200{,}000$ ...

Vô Pseudonym

29

asked2 days ago

Question 2

1vote

1answer

96views

Restricted Euler function

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 ...

Bruno Andrades

5,442

askedNov 20 at 18:05

Question 3

0votes

0answers

97views

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

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?...

Augusto Santi

3,032

askedNov 10 at 18:03

Question 4

14votes

2answers

717views

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

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$$...

Augusto Santi

3,032

askedNov 8 at 21:02

Question 5

3votes

1answer

111views

Connections between similar looking theorems

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 ...

abrahimladha

167

askedOct 13 at 15:04

Question 6

0votes

0answers

51views

How to solve the system involving factorials? [duplicate]

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 ...

Danilo Jonić

373

askedOct 8 at 20:47

Question 7

1vote

0answers

68views

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

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}\...

Mats Granvik

7,644

askedOct 6 at 11:10

Question 8

4votes

0answers

189views

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

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 ...

pi in the sky

41

askedOct 4 at 13:47

Question 9

0votes

0answers

47views

Common element with order constraints

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 ...

Turbo

6,341

askedSep 4 at 20:45

Question 10

1vote

1answer

87views

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

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 ...

michaelc35

221

askedAug 14 at 7:22

Question 11

0votes

1answer

76views

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

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 \...

Engineer

67

askedJul 23 at 5:53

Question 12

-3votes

1answer

200views

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

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 $\...

pie

9,247

askedJul 16 at 23:42

Question 13

1vote

1answer

103views

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

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)...

George Bailey

225

askedJul 9 at 22:01

Question 14

6votes

1answer

118views

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

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_{...

Faoler

2,854

askedJul 7 at 21:00

Question 15

3votes

3answers

258views

Do the set of coprimes of a number form an additive basis?

For any positive integer $n,$ define $X:= \{ a: (a,n) = 1,\ 1\leq a < n \}.$ Let $(X+X)^*:= \{ (x+y)\pmod n: x,y\in X \}.$Conjecture: If $n$ is even, then $(X+X)^*= \{ 0,2,4,\ldots,n-2\}.$ If $n$ ...

Adam Rubinson

25.2k

askedJun 27 at 14:09

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Stack Exchange Network

Questions tagged [totient-function]

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?

Do the set of coprimes of a number form an additive basis?

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