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from the wiki page:

"Zeckendorf's theorem states 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."

so, distinct and non-consecutive

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Non consecutive isn't surprising, any consecutive Fibonacci numbers in the sum can be replaced with their sum, which is itself a Fibonacci number by definition

That's not sufficient to be unsurprising.

What if a sum has more than 2 consecutive Fibonacci numbers? That doesn't cause a problem, but it takes a little more work to sort it out.

I find the distinct property much more interesting and non obvious.

It looks like non consecutivity is only there to force unique solutions hence called zeckendorf representations

Sure, but the conversion method does not require the numbers to be distinct or non-consecutive.

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