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 $n$ is an odd positive integer, then $2^{n!} − 1$ is divisible by $n$

Prove that if $n$ is an odd positive integer, then $2^{n!} − 1$ is divisible by $n$. Progress so far: Let $n=2k+1$. The desired result becomes $2^{(2k+1)!} − 1$ By Euler's totient function theorem, we have that $2^{\phi(2k+1)}=1\mod(2k+1).$ I…
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Eulers-phi function - calculating the last digit of a unending exponent

how can i calculate the last digit of the unending exponent 13^(13^(13.....) using Euler phi function? like to what modulus do i need to calculate the Euler phi function in order to calculate this. I know how to use this with a fixed exponent but…
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Arithmetic mean of the integers in set $S=\{k:k\in \mathbb{Z}, 1\leq k\leq n$ and $gcd(k,n)=1\}$

Or stated simply, what is the arithmetic mean of the totatives of $n$? From this question here I can see that the sum of the totatives is given by the formula $\large\frac{n\times\phi (n)}{2}\large$. So the required answer would then be…
StubbornAtom
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How do I find all n values for which the equation $\phi (n) = 8$ holds?

I've heard all kinds of different ways to solve this problem, yet haven't been able to apply them specifically to the number 8 (Worked fine for 6 for example). I'd love to see a well-explained solution, is possible. Thank you.
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Dealing with phi function property

If $n=2^kN$, where $N$ is odd, then $$\sum_{d\mid n}(-1)^{n/d}\phi(d)=\sum_{d\mid 2^{k-1}N}\phi(d)-\sum_{d\mid N}\phi(2^kd)$$ I have no idea how to seperate things inside the left side. In a nornal sum function it would be easy but any attempt…
Stav Alfi
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Calculate $a^8 \bmod 15$ for $a = 1,2,\dots,14$

I am trying to calculate $a^8 \bmod 15$ for $a = 1,2,\dots,14$ I get that because $a = 2,4,7,8,11,13,14$ are relatively prime to $15$, the answer will be $1$ in those cases. But how to get this for the other values of $a$?
devcoder
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How can you find $n$, such that $ \varphi(n) = \frac{2n}{5} $ where $\varphi$ is the totient function?

How can you find $n$, such that $$ \varphi(n) = \frac{2n}{5} $$ where $\varphi$ is the totient function? So let $n=2^a5^b$, then we get $$ \varphi(2^a5^b)=\varphi(2^a)\varphi(5^b)=2^{a-1}(2-1)5^{b-1}(5-1)=2^{a+1}5^{b-1}= 2n/5 $$ so every $n$ of…
calculatormathematical
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What is the remainder of $2^{4000}$ divided by 99?

Can someone guide me on how to find solution to such problems within a minute as that is the amount of time I will be given during my exams. also share what answer you get as I got different answers for same question with different methods
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Euler's totient function of $n^2$

Prove that $\varphi(n^2)=n\cdot\varphi(n)$ for $n\in \Bbb{N}$, where $\varphi$ is Euler's totient function.
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Finding expression for $\sum\limits_{n\leq x}\frac{\varphi(n)}{n^\alpha}$.

In Apostol's book on number theory, there is a problem related to average order of arithmetic functions: For $x\geq 2 $ show that, $\sum\limits_{n\leq…
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Prove that $ 2^{6ℓ + 2} ≡ 4\pmod{18}$

Prove that $2^{6ℓ + 2}≡ 4 \pmod{18}$ for $0\leℓ ∈ ℤ$.
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Is Euler's totient function surjective?

I other words can I find a number m for any number $n$ such that $\varphi(m)=n$? It would be great if you could also present a proof or a link to a paper that contains a proof Edited note: I forgot to mention $n$ must be even as $\varphi(m)$ is…
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how to calculate $\phi (625)$? Eulers's Totient

euler's totient relies on primes, and coprimes in order to determine $\phi (n)$ but 625 is not the product of any 2 primes, and none of its factors are coprimes so how would you determine $\phi (625)$?
Skrrrrrtttt
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Why is this sequence not in the Online Encyclopedia of Integer Sequences?

Why is this sequence $1, 30, 105, 248, 264, 714, \ldots$ not in OEIS? I got that sequence from this ProofWiki link. I also did some further searching via Google and found these two papers in the JOURNAL OF INTEGER SEQUENCES: On the Ratio of the…
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is there an elementary proof of $\sum\limits_{d|n}\phi(d)=n$?

this question has been asked for several times but I need an elementary solution without advanced techniques there are some links that can help you 1-$\sum_{d|n}\phi(d)=n$ 2-$\sum_{d|n}\phi(d)=n$ 3-$\sum_{d|n}\phi(d)=n$ 4-$\sum_{d|n}\phi(d)=n$
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