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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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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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.
Yami Kanashi
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