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Questions tagged [carmichael-function]

For questions on Carmichael functions.

In number theory, the Carmichael function of a positive integer $n$, denoted ${\displaystyle \lambda (n)}$, is defined as the smallest positive integer $m$ such that $a^m \equiv 1\pmod n$ for every integer a that is coprime to n. In more algebraic terms, it defines the exponent of the multiplicative group of integers modulo n. The Carmichael function is also known as the reduced totient function or the least universal exponent function, and is sometimes also denoted ${\displaystyle \psi (n)}$.

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Do $\lambda(n)$ and $\pi(n)$ coincide infinitely often?

Let $\pi(n)$ be the prime-counting function and $\lambda(n)$ the Carmichael-function. Does $$\pi(n)=\lambda(n)$$ hold for infinite many positive integers $n$ ? I have no idea for an approach other than just brute force. The solutions I got so far…
Peter
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The maximal size of between $\varphi(n)$ divided by $\lambda(n)$.

I want to find $$f(n) = \max\left\{\frac{\varphi(k)}{\lambda(k)} : 1 \leq k \leq n\right\}$$ In other words, I want to find the maximal value of $\frac{\varphi(k)}{\lambda(k)}$ when $k$ is restricted. $\lambda(n)$ is the Carmichael function, the…
wythagoras
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Does the fraction of positive integers not being a Carmichael value have a limit?

Let $f(n)$ be the number of positive integers $x\le n$ such that $\lambda(k)=x$ has no solution, where $\lambda(k)$ denotes the Carmichael-function. Does $$\lim_{n\rightarrow \infty} \frac{f(n)}{n}$$ exist , and if yes, is it $1$ or some smaller…
Peter
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Carmichael function and the largest multiplicative order modulo n

By definition, the Carmichael function maps a positive integer $n$ to the smallest positive integer $t$ such that $a^t\equiv1\pmod n$ for all integers $a$ with $\gcd(a,n)=1$. It is denoted as $\lambda(n)$. The Wikipedia page on Carmichael function…
Iming Cheng
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Conjecture about the Carmichael function

Let $\ \lambda(n)\ $ denote the Carmichael-function and define $\ n(e)\ $ to be the number of solutions of $\ \lambda(m)=e\ $. For $\ e\ge 3\ $ , we have $\ 4\mid n(e)\ $ I chose "$\ e\ $" because we can concentrate on then even numbers because…
Peter
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A346587: which numbers maximize $\frac{n}{\lambda(n)}$?

This is the sequence of $n$ so that $\frac{n}{\lambda(n)}$ is greatest up until $n$, where $\lambda$ is the Carmichael function. The analogous quantity $\frac{n}{\varphi(n)}$ is maximized when $n$ is a primorial, but as you can see in the link, the…
Derivative
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Addition to previously asked question on Generalized Carmichael Numbers

I had previously asked a question on mathstack exchange (Conjecture on The Generalized Carmichael Numbers) concerning with a conjecture I had discovered. I worked on the problem for a long time and discovered one more conjecture. With the same terms…
BookWick
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What is the state of Carmichael's totient function conjecture?

I have been searching for information about that conjecture and it seems for me that noone has made any significant improvement on it in the last 30 years. Is that true? Does it remain unproven to be true? Has there been any important discovery…
user3141592
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Implication related to carmichael function.

If $g \in \Bbb Z_{n^2}^{*}$ and $x_1,x_2 \in \Bbb Z_n$ then help me in proving the following implication. $g^{n \lambda(n)}\equiv 1 \mod{n^2} \implies g^{(x_1-x_2)\lambda(n)} \equiv 1 \mod{n^2}$ where $\lambda(n)$ is carmichael function. I know how…
hanugm
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Carmichael function available in PARI / GP?

Is the Carmichael function $\lambda(n)$ available in PARI / GP or do I have to program it ? I know the command znorder, but this does not seem to be enough to calculate the carmichael function. I have programmed the function, but it would…
Peter
  • 86,576
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Conjecture on The Generalized Carmichael Numbers

Define $$C_k = \{n \in \mathbb{Z}: n > \max(k, 0) \text{ }\text{and}\text{ }a^{n - k + 1} \equiv a \pmod{n} \text{ }\text{for all} \text{ } a \in \mathbb{Z}\}$$ Thus it's easy to see that when $k = 1$, $C_1$ consists of all of the prime numbers and…
BookWick
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wondering about multiplicative (not arithmetic) sequences of primes

(Apologies in advance if the terminology is wrong). I've been led by my research into looking at sequences of primes of the form $(p_1,p_2,\ldots,p_m)$ with each $p_i$ of the form $k_i(p_1-1)+1$, where $k_i$ is a strictly increasing finite sequence…
Barry Fagin
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the least $m$ such that $a^m\equiv 1 \mod n $ for fixed $a,n$.

Is there any known method for calculating $\lambda_a(n)$ which returns the smallest integer $m$ such that $a^m\equiv 1 \pmod n$ where $\gcd(a,n)=1$ ? I searched but I found nothing, is there at least an algorithm that does not use bruteforce ?…
user55386
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Relationship between the Carmichael function and Euler's totient function

Let $\lambda$ denote the Carmichael function and $\varphi$ Euler's totient function. Furthermore, let $p$ denote any prime number and $k\in\mathbb{N}$. The wikipedia article about $\lambda$ states: $$\lambda(p^k)=\begin{cases}\frac{1}{2}\varphi…
0xbadf00d
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On Carmichael function and aliquot parts of odd perfect numbers

This post is cross-posted on MathOverflow with identifier 439563 and same title. We denote as $N$ an odd perfect number, and $d\mid N$ one of its divisors. We denote the Carmichael function as $\lambda(x)$, Wikipedia has the article Carmichael…
user759001
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