Questions tagged [carmichael-function]
For questions on Carmichael functions.
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Why order of $a$ co-prime to $n$ traverses all factors of carmichael function of $n$? -Is it true?
I'm recently a bit indulged with basic number theory and investigating in the Carmichael $\lambda(n)$ function, which is defined as the smallest positive number that makes $a^x\equiv\ (\mathrm{mod}\ n)...
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Finding a General Explicit Formula for $\lambda(n)$ for when $n\neq p^k$
I have been trying to answer the following question: Is there a general formula for Carmichael's totient function ($\lambda(n)$) that is analogous to the Euler product formula for Euler's totient ...
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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 $\...
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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:...
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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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Carmichael function and the largest multiplicative order modulo n [duplicate]
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(...
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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 ...
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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 $$\lim_{n\rightarrow \infty} \frac{f(n)}{n}$$ ...
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Why is $\ 100\ $ so special in this context?
Let $\ \lambda(k)\ $ be the Carmichael function and denote $\ N(n)\ $ to be the number of solutions of $\ \lambda(k)=n\ $
Can we prove that $\ N(n)=n\ $ only holds for $\ n=100\ $ (besides the ...
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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\ $" ...
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Can the Carmichael Function be used with Proof By Infinite Descent in the following way?
I was reading up on the Carmichael Function and I had a question about its use.
Can it be used to show that if $x > 1$, then $n=1$ for:
$$2^m(2^x - 1) = 3^n(3^y - 1)$$
Here's the argument that I ...
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When is $\lambda (n)$ = $\phi (n)$?
$\lambda (n)$ is the Carmichael function and $\phi (n)$ is the Euler totient function. I can see that if $n$ is prime then the two functions agree and also if $n$ is a power of a prime they agree. Are ...
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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 ...
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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 = ...
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