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Let's say I have a number system with an array defining the bases at each position. For example, [1,1,1] would be a binary number system (each digit can be 0 or 1) in which any number from 0 to 3 can be represented. Similarly, I can have [3,2,1] where the units digit can be 0 or 1, the tens digit can be 0,1 or 2 and the hundreds digit can be 0,1,2 or 3. if these digits are $u$, $t$ and $h$ respectively the number becomes: $u+2t+3h$.

Let's define consistency as the property where every integer less than the max permissible value has one and only one representation in the base system.

Now, the second number system in the examples above isn't consistent. If I take 4, there are two ways to represent it: $[1,0,1] = 3*1+2*0+1*1$ or $[0,2,0] = 3*0+2*2+1*0$. On the other hand, [6,2,1] is consistent. Is there a way by looking at the array of bases weather or not the number system is consistent.

Rohit Pandey
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1 Answers1

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A result I proved about 50 years ago is that a representation $ n =\sum_{j=0}^{\infty} a_jB_j $ where the $a_j$ and $B_j$ are integers, $B_0=1$, $1 \le B_j < B_{j+1}$, and $0 \le a_j \lt \dfrac{B_{j}}{B_{j-1}} $ is unique and exists iff $\dfrac{B_{j+1}}{B_j}$ is an integer for all $j$.

This is why $B_j = m^j$ for integral $m\ge 2$ and $B_j = j!$ have a unique representation and $B_j = F_j$ (Fibonacci numbers) does not.

The representation or $n$ is gotten by the standard method of starting with the largest $B_k \le n$, setting $n_k = n$, $a_k =\lfloor \dfrac{n_k}{B_k} \rfloor $, $n_{k-1} =n_k-a_kB_k$, and iterating with $k = k-1$ until $k = 0$.

marty cohen
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