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Number Theory

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Universal Exponent Function: Which Statements Are True?

NUMTH-5EIZXY

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

If $9|\lambda(n)$, then the set of distinct prime divisors of $\lambda(n)$ and the set of distinct prime divisors of $\lambda(n)/3$ are identical.

B

Any natural number $n$ satisfying $\lambda(n)=\phi(n)$, is $n=1$, $2$, $4$, or $n=p^e$ for odd prime $p$ and $e\geq 1$.

C

For natural numbers $n$, $a$, $b$, and $c$ with $(a,n)=1$, $a^b\equiv a^c\ \pmod n \ $ if and only if $\ b\equiv c\ \pmod {\lambda(n)}$.

D

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

E

$3|\lambda(n)$ if and only if a prime $p\equiv 1 \ \textrm{mod} \ 3$ divides $n$.