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Fibo again (Posted on 2014-06-26) Difficulty: 2 of 5
Prove the following theorem:

Every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers without including any two consecutive Fibonacci numbers.

Examples: 8=8; 27=21+5+1

No Solution Yet Submitted by Ady TZIDON    
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Solution Solution Comment 1 of 1

First we observe that 1, 2 and 3 can all be represented as themselves; 4 = 3 + 1; and 5 is once again a Fibonacci number which can be represented as itself.  

Consider a Fibonacci number f_n.  Assume that all positive integers <= f_n can be represented as the sum of one or more distinct Fibonacci numbers (without including any two consecutive). 

Then f_n + 1, f_n + 2, f_n + 3, ..., f_n + f_(n-1) - 1 can all be represented as the sum of distinct Fibonacci numbers.  f_n + f_(n-1) would be the first case that would appear to violate the condition that there are no two consecutive in the sum, but f_n + f_(n-1) is simply f_(n+1) and can therefore be represented as itself. 

Thus, all positive integers can be represented in this way.


  Posted by tomarken on 2014-06-26 09:53:47
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