I have questions about (and am looking for information on) the powers of $2$ modulo $b$, where $b$ is an odd composite number.
Specifically:
1) I know that $2^{\lambda\left(b\right)}\overset{b}{\equiv}1$ (where $\lambda$ is the Carmichael function). Is there anything that can be said about the value of the (multiplicative) order of $2$ mod $b$ beyond the fact that it divides $\lambda\left(b\right)$? Are there any meaningful controls, formulae, or estimates relating the value of $\lambda\left(b\right)$ to the order of $2$ mod $b$?
2) Given an integer $a$, I'll write $\left\langle 2\right\rangle _{a}$ to denote the multiplicative subgroup of $\mathbb{Z}/a\mathbb{Z}$ generated by $2$. Letting $p_{1},\ldots,p_{N}$ be distinct odd primes listed in increasing order, and letting $e_{1},\ldots,e_{N}$ be positive integers, are either of the following formulae for the order of $\left|\left\langle 2\right\rangle _{a}\right|$ even remotely close to being true:$$\left|\left\langle 2\right\rangle _{p_{1}^{e_{1}}\cdots p_{N}^{e_{N}}}\right|\overset{?}{=}\left|\left\langle 2\right\rangle _{\max\left\{ p_{1}^{e_{1}},\ldots,p_{N}^{e_{N}}\right\} }\right| $$ or possibly: $$\left|\left\langle 2\right\rangle _{p_{1}^{e_{1}}\cdots p_{N}^{e_{N}}}\right|\overset{?}{=}\max\left\{ \left|\left\langle 2\right\rangle _{p_{1}^{e_{1}}}\right|,\ldots,\left|\left\langle 2\right\rangle _{p_{N}^{e_{N}}}\right|\right\}$$
If so, why? If not, is there anything along these lines that can be said about the multiplicative order of $2$?
3) It is known that the multiplicative group of integers modulo $b$ is cyclic if and only if $b$ is one of the numbers $\left\{ 1,2,4,p^{k},2p^{k}\right\}$, where $k$ is a positive integer and $p$is an odd prime. Since I need $b$ to be odd, this only leaves the prime power groups $p^{k}$. As such: is there anything known about the relation between $\left|\left\langle 2\right\rangle _{p^{k}}\right|$, the odd prime $p$, and the integer $k$? One pattern that I have noticed is that the action of $2$ (by repeated multiplication) on the set $p^{j}$ (for $1\leq j\leq k-1$) has an order/period of $\left|\left\langle 2\right\rangle _{p^{k-j}}\right|$.
Example (mod $3^{3}$):
$$\left\langle 2\right\rangle _{3^{3}} = \left\{ 2,4,8,16,5,10,20,13,26,25,23,19,11,22,17,7,14,1\right\}$$ $$2:3 : \left\{ 3,6,12,24,21,15\right\}$$ $$2:3^{2} : \left\{ 9,18\right\}$$ Here, $\left|\left\langle 2\right\rangle _{3^{2}}\right|=6$, and $\left|\left\langle 2\right\rangle _{3}\right|=2$, which are, clearly, the orders of the actions of $2$ on $3^{1}$ and $3^{2}$, respectively.
Any answers—and relevant resources in the literature—would be much appreciated.