# Questions tagged [carmichael-function]

For questions on Carmichael functions.

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### Alternative definition for carmichael function

As my textbook describes it, the Carmichael function is the minimum integer m such that, for all a coprime with n, $$a^m \equiv 1 \pmod{n}$$ So, as far as I understand it, m is the minimum number that:...
1 vote
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### When does a solution of $m+\lambda(m)=n$ exist?

Let $\lambda(m)$ be the Carmichael-function. For which positive integers $n$ does a positive integer $m$ exist with $m+\lambda(m)=n$ ? In other words, for which $n$ is $m+\lambda(m)=n$ solvable ? Of ...
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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 ...
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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 $k_i(p_1-1)+1$, ...
1 vote
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### Clarification about Carmichael's Lambda Function

By definition Carmichael function $\lambda(n)$ is the the smallest positive integer $m$ such that $$x^m\equiv 1\pmod{n}$$ for all $1\leq x\leq n$ such that $\gcd(x,n)=1$. Moreover it is simple to ...
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1 vote
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### Are these limits correct? $\lim_{n\to \infty}\text{sup} \frac{\lambda (n)}{n}=1$ and $\lim_{n\to \infty}\text{inf} \frac{\lambda (n)}{n}=0$ exist?

I learned that from here for Euler totient function $\phi (n)$ , we have $$\lim_{n\to \infty}\text{sup} \frac{\phi (n)}{n}=1$$ $$\lim_{n\to \infty}\text{inf} \frac{\phi (n)}{n}=0$$ However, I could ...
1 vote
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### How is Carmichael's function subgroup of Euler's Totient function?

I've seen this question. I'm trying to find the connection between Euler's totient function and Carmichael's function. Carmichael's function outputs smallest $k$ such that: $a^k ≡ 1 \pmod n$ Euler'...
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### Is the order of integers mod p under multiplication output of Carmichael's function?

I've been trying to understand the group theory proof of Fermat's little theorem. Let's say there exists group $G = (ℤ/pℤ)^x$, and it has multiplicative subgroup $H$ (a monoid group, as i understand)...
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### There are infinitely many $n$ such that $\lambda(n) = k$?

There are infinitely many $n$ such that $\lambda(n) = k$ (Carmichael function)? For example: $k = 4$. How efficiently we can generate all $n$ for which $\lambda(n) = 4$?
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### Is there any way to find a number N if it's Carmichael function is given.

I know to find the Carmichael function [ C() ] of a given no. But I want to know if there is any method or shortcut to find a number N if it's C() is given.
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### Shortest repeating digits in fractions.

If you have an integer of $n$ decimal digits that is odd and non-divisble by $5$ then what is the shortest repeating decimal it can have in its reciprocal? Can it be as small as $n^c$ digits for some ...
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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 ...
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1 vote
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### Suppose that $n$ is a composite, squarefree integer such that for every prime divisor $p$ of $n$... [duplicate]

Suppose that $n$ is a composite, squarefree integer such that for every prime divisor $p$ of $n$, we have $(p - 1) | (n - 1)$. Prove that $n$ is a Carmichael number. Having a lot of trouble with this ...
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### For which numbers, every possible order can be achieved?

Let $n\ge 2$ be a natural number. Then, every possible order of an number modulo $n$ is a divisor of $\lambda(n)$. $\lambda(n)$ is the Carmichael-function, it is the largest possible order modulo $n$....
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1 vote
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### What is the largest number $n$ with $\lambda(n)=k$ for a given $k$?

Let $k\ge 2$ be a natural number. What is the largest number $n$ with $\lambda(n)=k$, where $\lambda(n)$ is the Carmichael-function. (See : https://en.wikipedia.org/wiki/Carmichael_function ) ? ...
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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 ...
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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....
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1 vote
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### Carmichael function and primitive roots of unity

I have been reading about the Carmichael function recently and I would like to ask about some elementary implication of its properties as I haven't found it stated explicitly. If I understand it ...
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### How many Fermat tests are needed to verify a Carmichael number

If $n$ is a Carmichael number, then for all values $a$ such that $0<a<n$ (and $a \perp n$): $a^{n-1} \equiv 1 \mod n$ However, is it not necessary to check check all $a$ values because for a ...
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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. $g^{n \lambda(n)}\equiv 1 \mod{n^2} \implies g^{(x_1-x_2)\lambda(n)} \equiv 1 \mod{n^2}$ ...
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