?

Free Version

Upgrade subject to access all content

Moderate

Universal Exponent Function

NUMTH-IEGQSV

The lambda function $\lambda(n)$ is defined as the smallest positive integer $m$ such that

$$a^m \equiv 1 \pmod{n}$$

for every integer $a$ that is coprime to $n$. We say that $\lambda(n)$ is the universal exponent of the group $(\mathbb{Z}/n\mathbb{Z})^{*}$. The lambda function can be evaluated by the following procedure:

$$ \lambda(n)=\textrm{l.c.m.} (\lambda(p_1^{e_1}), \cdots, \lambda(p_r^{e_r}))$$

...where

$$n=p_1^{e_1}\cdots p_r^{e_r} \ \ (\mbox{$p_i$ 's distinct prime numbers}),$$
$$ \lambda(p^e)=p^{e-1}(p-1) \ \ \mbox{ if $p$ is odd prime, and $e\geq 1$,}$$
$$ \lambda(2^e)=2^{e-1} \ \ \mbox{ if $1\leq e\leq 2$,}$$
$$ \lambda(2^e)=2^{e-2} \ \ \mbox{ if $e\geq 3$, and}$$
$$ \lambda(1)=1. $$

Which of the following statements are true?

Check ALL that apply.

A

$\lambda(n)=\phi(n)$ for infinitely many $n$.

B

$\lambda(2n)=\lambda(n)$ for any $n$

C

$\lambda(n)$ is a multiplicative order of some element $a\in(\mathbb{Z}/n\mathbb{Z})^{*}$.

D

$\lambda(n^2)=\lambda(n)$ for infinitely many $n$.

E

$\lambda(mn)=\lambda(m)\lambda(n)$ for coprime $m$, $n$.