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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If $a^\phi = 1$ then the order of $a$ divides $\phi$

How can I show that the order of an element modulo $m$ divides $\phi(m)$? I know that if $a$ and $m$ are relatively prime, then the least positive integer $x$ such that $a^x\equiv1\pmod m$ is its order modulo $m$. I also know that, by Euler's…
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Euler's Totient function $\forall n\ge3$, if $(\frac{\varphi(n)}{2}+1)\ \mid\ n\ $ then $\frac{\varphi(n)}{2}+1$ is prime

While I was studying Euler's Totient function, $\varphi(n)$, I stumbled upon the marvelous book "Index to Mathematical Problems, 1980-1984" By Stanley Rabinowitz. In this page of the book (link to Google books sample), the open problems and…
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$\phi(\pi)$ and other irrationals (Euler's totient function)

Over the natural numbers, Euler's totient function $\phi(n)$ has the nice property that $\phi(n^m)=n^{m-1}\phi(n)$. I've found that this can naively extend the totient function over the rationals…
Graviton
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For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there exists infinitely many $x$ for…
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Very elementary proof of that Euler's totient function is multiplicative

Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function. My problem is this: I'd like to explain this to some gifted children. The children are gifted enough to…
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How to calculate these totient summation sums efficiently?

I am trying to find good ways to tackle sums of the form $\sum_{k=1}^{N}k^j\varphi(k)$ $j$ can be anything but I am largely concerned about cases 0, 1, and 2. $\varphi(k)$ is the Euler totient function. Can this be done without needing to calculate…
Sean Hill
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Euler totient function sum of divisors. Theorem 2.2 Apostol

Prove that : If $ n\ge 1 $, then $ \sum_{d|n}\phi(d)=n $. Let $S$ denote the set $\{1,2,...,n\}$. We distribute the integers of $S$ into disjoint sets as follows. For each divisor $d$ of $n$, let $A(d) = \{k \in S :(k,n) = d\}$ That is,…
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Why is $f(x) = x\phi(x)$ one-to-one?

I noticed that $f(x) = x\phi(x)$ seems to be one-to-one, where $\phi(x)$ is Euler's Phi function. In particular, I'm writing some numerical python code and the line I have looks something like sorted([n*phi(n) for n in range(1,1000)]) and there are…
James
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inversion of the Euler totient function

Given an integer $n$ find smallest integer $x$ such that $\varphi(x) = n$. $$10^5 < n < 10^8$$ I know that lower bound for searching is $n+1$ and the upper bound is $$\frac{n}{e^{0.577}\log(\log(n))} + \frac{3}{\log(\log(n))}$$ But my problem is…
Purva Gupta
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Prove that $\phi(n) \geq \sqrt{n}/2$

So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$.…
Numbersandsoon
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On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $

There is a second part of the problem posted in Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$, from Apostol's book, but I can't figure it out. It asks the reader to prove that if $x \geq 2$ then…
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Finite-order elements of $\text{GL}_4(\mathbb{Q})$

I'm currently studying for my qualifying exams in algebra, and I have not been able to solve the following problem: Determine all possible positive integers $n$ such that there exists an element in $\text{GL}_4(\mathbb{Q})$ of order $n$. I've been…
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A conjecture: for all $n\in\mathbb{N}$, the least $k>1$ such that $\phi(k)\geqslant n$ is a prime

I came across a problem in a book that asked us to find the first number $n$ such that $\phi(n)\geqslant 1,000$ it turns out that the answer is 1009, which is a prime number. There were several questions of this type, and our professor conjectured…
JimmyJackson
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An integer is prime iff $\phi(n) \mid n-1$ and $n+1 \mid \sigma (n)$

I wish to prove An integer is prime iff $\phi(n) | n-1$ and $n+1|\sigma (n)$ where $\phi$ is Euler's totient function and $\sigma(n)$ is the sum of the positive divisors of n. I can show from a previous exercise that $\phi(n)|n-1$ implies n is…
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