Representation of Integers in Balanced Ternary

Theorem
Let $n \in \Z$ be an integer.

$n$ can be represented uniquely in balanced ternary:


 * $\ds n = \sum_{j \mathop = 0}^m r_j 3^j$


 * $\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}$

such that:

where:
 * $m \in \Z_{>0}$ is a strictly positive integer such that $3^m < \size {2 n} < 3^{m + 1}$
 * all the $r_j$ are such that $r_j \in \set {\underline 1, 0, 1}$, where $\underline 1 := -1$.

Proof
Let $n \in \Z$.

Let $m \in \Z_{\ge 0}$ be such that:
 * $3^m + 1 \le \size {2 n} \le 3^{m + 1} - 1$

where $\size {2 n}$ denotes the absolute value of $2 n$.

As $2 n$ is even, this is always possible, because $3^r$ is always an odd integer for non-negative $r$.

Let $d = \dfrac {3^{m + 1} - 1} 2$.

Let $k = n + d$.

We have that:

Let $k = n + d \in \Z$ be represented in ternary notation:
 * $k = \ds \sum_{j \mathop = 0}^m s_j 3^j$

where $s_j \in \set {0, 1, 2}$.

By the Basis Representation Theorem, this expression for $k$ is unique.

Now we have:

Hence we see:

Hence $n$ has a representation in balanced ternary.

The representation for $k$ in ternary notation is unique, as established.

Hence the representation in balanced ternary for $n$ is also unique.