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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Pythagorean triples that "survive" Euler's totient function

Suppose you have three positive integers $a, b, c$ that form a Pythagorean triple: \begin{equation} a^2 + b^2 = c^2. \tag{1}\label{1} \end{equation} Additionally, suppose that when you apply Euler's totient function to each term, the equation…
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All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have $\phi(n) \mid \phi(P(n))$. It is conjectured there…
Amir Parvardi
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What's the proof that the Euler totient function is multiplicative?

That is, why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers? It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation…
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Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$ I have…
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Prove $n\mid \phi(2^n-1)$

If $2^p-1$ is a prime, (thus $p$ is a prime, too) then $p\mid 2^p-2=\phi(2^p-1).$ But I find $n\mid \phi(2^n-1)$ is always hold, no matter what $n$ is. Such as $4\mid \phi(2^4-1)=8.$ If we denote $a_n=\dfrac{\phi(2^n-1)}{n}$, then $a_n$ is A011260,…
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Why is Euler's Totient function always even?

I want to prove why $\phi(n)$ is even for $n>3$. So far I am attempting to split this into 2 cases. Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will always be even. Case 2: $n$ is not a power of $2$. This is…
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Does there exist $a_0$, such that $\{a_n\}_{n=0}^{\infty}$ is unbounded?

Suppose $\{a_n\}_{n=0}^{\infty}$ is a sequence, defined by the recurrence relation $$ a_{n+1} = \phi(a_n) + \sigma(a_n) - a_n, $$ where $\sigma$ denotes the divisor sum function and $\phi$ is Euler's totient function. Does there exist $a_0$ such…
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Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?

If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically partitions $\mathbb{Z}/n\mathbb{Z}$ into subsets of…
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Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. While this is a perfectly good proof, I have to…
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Iterated Euler's totient function

Let $\phi(n)$ be the Euler totient function: $$ \phi(2)=1 \;,\; \phi(11)=10 \;,\; \phi(12)=4\;,$$ etc. Define $\Phi(n)$ to be the number of iterations $k$ so that $\phi^k(n)$ reaches $1$. For example, $\Phi(25)=5$ because $\phi(25)=20$ and…
Joseph O'Rourke
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Is the Euler phi function bounded below?

I am working on a question for my number theory class that asks: Prove that for every integer $n \geq 1$, $\phi(n) \geq \frac{\sqrt{n}}{\sqrt{2}}$. However, I was searching around Google, and on various websites I have found people explaining that…
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Proof of a formula involving Euler's totient function: $\varphi (mn) = \varphi (m) \varphi (n) \cdot \frac{d}{\varphi (d)}$

The third formula on the wikipedia page for the Totient function states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$ where $d = \gcd(m,n)$. How is this claim justified? Would we have to use the Chinese Remainder…
The Chaz 2.0
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New identity for Euler's Totient Function?

A few weeks ago I discovered and proved a simple identity for Euler's totient function. I figured that someone would have already discovered it, but I haven't been able to find it anywhere. So I was hoping someone would be able to tell me whether…
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How prove that:$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\frac{n(n-1)}{4}+1$

Prove that for $n\ge 3$, $$\varphi(2)+\varphi(3)+\varphi(4)+\cdots+\varphi(n)\ge\dfrac{n(n-1)}{4}+1$$ where $\varphi$ is the Euler's totient function I think we must use…
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Prove that there are no composite integers $n=am+1$ such that $m \ | \ \phi(n)$

Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a
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