Goldbach's Conjecture and Zeckendorf's Theorem

Watching the movie A Brilliant Young Mind a couple of nights ago, I heard mention made of Goldbach's Conjecture and thought I should find out about it. This is the introduction to what Wikipedia had to say about it: 

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to hold up through 4 × 10^18, but remains unproven despite considerable effort.

The expression is not unique as the example of 100 shows:

100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53

I came across Zeckendorf's Theorem when examining the number 24331. It's similar to Goldbach's Conjecture in that it deals with the decomposition of numbers but into Fibonacci numbers, not primes. It states, to quote from Wikipedia again, that:

Every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers.

Now the first few Fibonacci numbers are:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368

The Wikipedia article goes on to state that the theorem has two parts:

Existence: every positive integer n has a Zeckendorf representation.
Uniqueness: no positive integer n has two different Zeckendorf representations.

The example is given of the number 100:

100 = 89 + 8 + 3.

There are other ways of representing 100 as the sum of Fibonacci numbers:

100 = 89 + 8 + 2 + 1

100 = 55 + 34 + 8 + 3

but these are not Zeckendorf representations because 1 and 2 are consecutive Fibonacci numbers, as are 34 and 55.