Questions tagged [totient-function]

Questions on the totient function $\phi(n)$ (sometimes $\varphi(n)$) of Euler, the function that counts the number of positive integers relatively prime to and less than or equal to $n$.

The Euler phi function $\phi(n)$ is defined to be the number of integers between $1$ and $n$ which are relatively prime to $n$; that is,

$$\phi(n) = |\{1 \le k < n : \gcd{(k, n)} = 1\}|$$

The phi function is multiplicative; that is, if $n$ and $m$ are relatively prime, then

$$\phi(nm) = \phi(n) \phi(m)$$

There is also a product representation,

$$\phi(n) = n \prod_{p|n} \left(1 - \frac{1}{p}\right)$$

where the product is taken over prime divisors of $n$.

One particularly important use of Euler's phi function is in computing exponents with modular arithmetic. Whenever $a$ and $n$ are relatively prime, Euler's theorem states that

$$a^{\phi(n)} \equiv 1 \pmod{n}$$

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Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?

Is it true that : $\frac {n}{5} < \phi (n) < n$ for all $n > 1$ where $\phi (n)$ is Euler's totient function . Since $\phi(n)$ has maximum value when $n$ is a prime it follows that maximum value of $\phi(n)$ in term of $n$ is $n-1$ , therefore…
Peđa
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What is a good way to introduce Euler's totient function?

I was thinking of this question and when I googled I couldn't find any MSE results, but I found one from Reddit. I just wanted to ask the question here and post the answer as community wiki just so MSE could have some discussion. If you want this…
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Modified Euler's Totient function for counting constellations in reduced residue systems

I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three parts: "background" describes how this function…
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Values taken by Euler's phi function

What are the values taken by Euler's phi function? For example, one can show that $\varphi(n)$ is even for $n \neq 1$ or $2$ by reducing the equality $\displaystyle n= \sum\limits_{d|n} \varphi(d)$ modulo $2$; so $\varphi (\mathbb{N}^*) \subset…
Seirios
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Are there infinite many solutions of $\ \ |\varphi(n+1)-\varphi(n)|=2\ \ $?

The solutions of the equation $$|\varphi(n+1)-\varphi(n)|=2$$ upto $\ \ n=10^8-1\ \ $ are (the first entries of the arrays) : ? j=1;a=[1,1];while(j<10^8,j=j+1;[a,b]=[b,eulerphi(j)];if(abs(a-b)==2,print([j-1 ,j,a,b]))) [4, 5, 2, 4] [5, 6, 4, 2] [7,…
Peter
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totient function series diverges?

Earlier today I thought I proved that the following series diverged: $$\sum_{n=2}^\infty\frac{\phi(n)}{n^2}$$ as a result of a misapplication of the prime number theorem. I mistook $\phi(n)$ for $\pi(n)$ in the statement. Is this salvageable? I've…
Prototank
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A possible Property of Euler's totient function: $n$ such that $\varphi(n)$ and $\varphi(n+1)$ are both powers of two

$n$ is an odd positive integer such that $\varphi(n)$ and $\varphi(n+1)$ are both powers of two . Here , $\varphi(n)$ denotes Euler's totient function. Is it true that $(n+1)$ is either $6$ or a power of $2$? Please help me to prove or disprove…
John
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Proving formula involving Euler's totient function

This question is motivated by lhf's comment here . "It'd be nice to relate this formula with the natural mapping $U_{mn} \to U_m \times U_n$ by proving that the kernel has size $d$ and the image has index $\varphi(d)$." Here, $U_k$ denotes the…
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Characteristic of a finite ring with $34$ units

Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the Euler totient, then $\varphi(k)$ divides $34$, hence…
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Does the following proposition hold in number theory

I am an undergraduate student in Bachelors in Mathematics undergoing a course in Algebraic Number Theory. I am stuck on the following problem: Let $n\in \mathbb N$. Let $d_1
Olivia
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Is the function $f(n)=\varphi(n)+\varphi(n+1)-n$ surjective?

For every positive integer $n$ define $$f(n)=\varphi(n)+\varphi(n+1)-n$$ $\varphi(n)$ denotes the totient-function. Is $f(n)$ surjective on the non-negative integers ? The first non-negative integer $k$ for which I yet did not find a positive…
Peter
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A congruence with the Euler's totient function and sum of divisors function

Can you provide a proof or a counterexample to the claim given below ? Inspired by the congruence $1.2$ in this paper I have formulated the following claim : Let $n$ be a natural number , let $\sigma(n)$ be sum of divisors function and let…
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Is my shorter expression for $ s_m(n)= 1^m+2^m+3^m+\cdots+(n-1)^m \pmod n$ true?

I'm considering the following sums for natural numbers n,m $$ s_m(n)= \sum_{k=1}^{n-1} k^m =1^m+2^m+3^m+\cdots+(n-1)^m $$ modulo n . Looking at odd n first, I found by analysis of the pattern of $s_m(n) \pmod n \quad $ the following expression…
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Updates on Lehmer's Totient Problem

As I read here and in many books on the Theory of Numbers, we are yet to prove or disprove the existence of any composite $n$ such that $\phi(n)\mid n-1$. Is there progress in this line?
lab bhattacharjee
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Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them

Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them. Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent…
Saurabh
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