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clarification on the Modular Arithmetic question by Borwein

In his answer to the previous Modular Arithmetic question What does the letter g mean in the $N^g$ bit sequence of a cubic residue? Jyrki Hukkinen writes:

[…] However, this sequence would not be a very efficient way of finding $g$. Why not?

I assume that this is a question about the matter of representation of numbers as bit sequences, which is the basis for the sequence $(N^{g_n})$, where $n=\log_2(N)$, and $g_n$ is the $n$-th digit of the $N^g$ bit sequence.
If $N$ is a number such that $N^g$ has a concise representation as a sequence of bits, then $N$ is also an integer power of $2$, which is the case in question here.
However, I don’t see why a sequence $(N^{g_n})$ cannot have a concise representation, so I just need to clarify a couple of things here.

Why is it important for the digit $g$ to be written in binary, and not for it to be written in decimal? Why can’t the number $g$ be represented as a real number?
In particular, let $\varphi$ be the Euler’s totient function, and let $\alpha$ be an integer. Then I’m assuming that the digit $g$ is written in binary, so it would mean that
$$\varphi(2^\alpha) = g\varphi(2^{\alpha-1})+g\varphi(2^{\alpha-2})+…+g = g\varphi(2^0)+g\varphi(2^1)+g\varphi(2^2)+…+g\varphi(2^\alpha-1)$$
But shouldn’t it be \$\varphi(2^\alpha)=g\varphi(2^{\alpha-1})+g\varphi(2^{\alpha-2})+…+g = (g+1)\varphi(2^
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