(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

prove that each nonzero integer may be uniquely represented in the form e_{0}+ e_{1}3^{1}+ e_{2}3^{2}+ ... + e_{k-1}e^{k-1}+ e_{k}3^{k}where e_{k}=/= 0 and each e_{k}= -1, 0, or 1.

2. Relevant equations

3. The attempt at a solution

I feel like this has to do with the basis representation theorem because it's in that section of my textbook. It looks to me like one would be able to represent the integers in base 3 since that's basically what it's saying but that would be only obvious for the positive integers and I'm not sure how to solve for that, or where the e_{k}= -1 would come in... any help would be appreciated, thanks!

[edit]

I did out for the first few integers, 1,...6 or so and found

if n=1 e_0 = 1 => 1 = 1

if n=2 e_0 = -1 and e_1 = 1 => -1 + 3(1) = 2

if n=3 e_0 = 0 and e_1 = 1 => 0 + 3(1) = 3

if n=4 e_0 = 1 and e_1 = 1 => 1 + 3(1) = 4

if n=5 e_0 = -1 and e_1 = -1 and e_2 = 1 => -1 + 3(-1) + 3*3(1) = 5

if n=6 e_0 = 0 and e_1 = -1 and e_2 = 1 => 0 + 3(-1) + 3*3(1) = 6

so there seems to be the pattern that e_0 goes 1, -1, 0 for each successive n and maybe e_1 goes 1,1,1,-1,-1,-1(?) for each successive n after n=1. I'm not sure where to go with this except that it seems to show something of a pattern, I just don't understand how to go about proving it.

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# Prove each nonzero integer may be uniquely represented

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