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totient

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totient
totient Math. (ˈtəʊʃənt) [irreg. f. L. totiēs, totiens, f. tot so many, after quotient.] The number of numbers (including unity) less than and prime to a given number. So totitive (ˈtɒtɪtɪv) [irreg. f. L. tot + -itive in such words as primitive, unitive], any one of such numbers in relation to the g... Oxford English Dictionary
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Highly totient number
A highly totient number is an integer that has more solutions to the equation , where is Euler's totient function, than any integer below it. , 34, 37, 38, 49, 54, and 72 totient solutions respectively. wikipedia.org
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Perfect totient number
, so 9 is a perfect totient number. Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. wikipedia.org
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Totient-like function I have number written as factors for instance: n = 2 * 3 * 3 * 5. What I have to do is find how many numbers between <1, n) are co-prime to n, which means GCD = 1. It can simply be done using Eul...
Let $n > 1$, and let $d < n$ be a positive divisor of $n$. You want to count the number of elements of the set $$ A = \\{ a : 0 \le a < n, \gcd(a, n) = d \\}. $$ Note that if $a \in A$, then $\gcd\left(\dfrac{a}{d}, \dfrac{n}{d}\right) = 1$, so $\dfrac{a}{d} \in B$, where $$ B = \left\\{ b : 0 \le b...
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Sparsely totient number
In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, n, is sparsely totient if for all m > n, where is Euler's totient function. wikipedia.org
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Are all values of the totient function for composite numbers between two consecutive primes smaller than the totient values of those primes? I've seen the graph for $\varphi(n)$ vs $n$ on Wolfram, and it seems like th...
Let's assume that Oppermann's conjecture is true and for every prime $p_n$ and next prime $p_{n+1}$,$$p_{n+1}p_n$$so your conjecture has counterexample.
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Is the totient function $\varphi$ invertible? As title, is the totient function $\varphi: \mathbb{N} \to \mathbb{N}$ invertible?
Note that $\varphi(1)=\varphi(2)$. More generally, $\varphi(2n)=\varphi(n)$ if $n$ is odd. So the $\varphi$-function is not one to one. It is also not onto. For if $b\gt 1$ is odd, there is no $n$ such that $\varphi(n)=b$. **Remark:** A fairly recent result of Ford shows that for any integer $k\ge 2...
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Euler's totient function
Sylvester coined the term totient for this function, so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Totient numbers A totient number is a value of Euler's totient function: that is, an for which there is at least one for which . wikipedia.org
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Jordan's totient function
In number theory, Jordan's totient function, denoted as , where is a positive integer, is a function of a positive integer, , that equals the number of generalization of Euler's totient function, which is the same as . wikipedia.org
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Divisibility property of the totient function $\varphi$ Let $n$ be a positive integer. Prove that \begin{equation} \sum_{d \vert n} \varphi(d)= n \end{equation} where $\varphi$ is the totient function of Euler.
For $d\vert n$, let $\mathcal{O}_d$ be the sets of elements of order $d$ in $\mathbb{Z}/n\mathbb{Z}$. Using Lagrange's theorem, $\\{\mathcal{O}_d\\}_{d\vert n}$ is a partition of $\mathbb{Z}/n\mathbb{Z}$. Howeover, as an element of order $d$ spans a group isomorphic to $\mathbb{Z}/d\mathbb{Z}$, $\ma...
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Carmichael's totient function conjecture
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number Examples The totient function φ(n) is equal to 2 when n is one of the three values 3, 4, and 6. wikipedia.org
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Totient function trick? I would like to search for primes of the form $$\varphi(n)^n+n$$ where $\ \varphi(n)\ $ denotes the totient function. The problem is that neither pfgw nor factordb seems to support the totien...
The totient function of an integer $n$ with prime factorisation $\prod \limits_{k=1}^r p_k^{\alpha_k}$ is given by $\varphi(n) = \prod \limits_{k=1}^r This allows you to determine the totient based off a prime factorisation, and furthermore also tells you that $n$ cannot have a square factor since if
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Lehmer's totient problem
In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. wikipedia.org
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Proving the Converse of Euler's Totient Theorem Euler's Totient Theorem states that if $n$ and $a$ are coprime positive integers, then $$a^{\varphi (n)} \equiv 1 \pmod{n}$$ Wikipedia claims that the converse of Eule...
Suppose that for some $B$, $a^B=1 \mod n$. Then we have some integers $k,m$ and an equation of the form $a^Bk+mn=1$. This means that $a^B$ and $n$ are coprime. Then $a$ and $n$ must be coprime.
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What is the Euler Totient of Zero? Wolfram MathWorld defines the Euler Totient function as follows: The totient function phi(n), also called Euler's totient function, is defined as the number of ...
Yes, because we must conclude that there are zero positive integers less than or equal to zero. We do not even have to consider the remainder of the criteria or the fact that $1$ is relatively prime to all numbers.
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