In this book that I read, the basis representation theorem is stated like this:
Let $k$ be any integer larger than $1$. Then, for each positive integer $n$, there exists a represenation $$n=a_0k^s+a_1k^{s-1}+...+a_s$$, where $a_0\neq 0$ and where each $a_i$ is nonnegative and less than $k$. Furthermore, this representation of $n$ is unique; it is called the representation of $n$ to the base $k$.
This statement is followed by a proof, and after a proof, there are some exercises.
One of them is worthwhile of stating here, it is:
Prove that each integer may be uniquely represented in the form $$n=\sum_{j=0}^{s}c_j3^j$$, where $c_s \neq 0$, and each $c_j$ is equal to $-1$,$0$, or $1$
This means that, if the base is $3$ then not only is the set $\{0,1,2\}$ a good choice for coefficients $a_j$ to be from that set, but is also the set $\{-1,0,1\}$.
Let us say that an integer $n \in \mathbb N$ is representable in base $b \in \mathbb N \setminus \{1\}$ if there exists natural number $s=s(n,b)$ such that $$n=\sum_{k=0}^{s}a_kb^k$$ and $a_j \in \{0,1,...b-1\}$ for $j=0,...,s$.
Basis representation theorem tells us that every $n \in \mathbb N$ is representable in every base $b \in \mathbb N \setminus \{1\}$, and not only that it is representable, but also uniquely.
Now, to be little more general, if we choose some natural number and a base $b$ and if we want that coefficients satisfy requirement $0\leq|a_j|<b$, so to allow that some of them can be negative then we have $2b-1 \choose b$ different possibilities for the set from which coefficients are to be.
Let us call a natural number $n$ a differently-representable in base $b>1$ if there exist at least two different coefficient-sets $\{d_0,...d_{b-1}\}$ and $\{e_0,...,e_{b-1}\}$ such that we have $0\leq d_i<|b|$ and $0\leq e_i<|b|$ for $i=0,...b-1$ and $$n=\sum_{j=0}^{s}e_jb^j$$ and $$n=\sum_{j=0}^{r}d_jb^j$$ for some $r,s \in \mathbb N_0$.
The basis representation theorem and this exercise tell us that every natural is differently-representable in base $3$.
I have two questions (hopefully, not too hard, and I do not know how to settle them):
1) Is there an infinite number of bases in which every natural number is differently-representable?
2) Is there an infinite number of bases in which every natural number is not differently-representable?
Perhaps it is true, by looking at the example with base $3$, that if the base $b=2w-1>1$ is an odd number, that then bases $\{0,1,...,2w-1\}$ and $\{-w+1,...,-1,0,1,...w-1\}$ are enough to consider to prove 1)?
We can do better yet. In any base $b\gt 2$ you can choose any number $k$ with $0 \le k \lt b-1$ and the digits $\{-k,-k+1,-k+2,\ldots b-1-k\}$ will suffice to represent any positive whole number.
Note that we can represent any number less than $b$ either as the single digit or as $1$ followed by a negative digit. We guaranteed that $1$ is available by forcing $k \lt b-1$. To represent a number $n$, write $n=bn'+r$ with $0 \le r \lt b$. Represent $r$ as in the first sentence. If $n'=0$ we are done. Otherwise, if $r$ is represented as a single digit represent $n'$ and append $r$. If $r$ is represented as $1$ followed by a negative digit, represent $n'+1$ and append the negative digit.
If you allow a negative sign, you can allow $k$ to be $b-1$, which makes base $2$ work as well. You just express numbers as usual in base $b$ but all the digits are negative so you put a minus sign in front to get positive numbers.