mersenneforum.org > Math Fibonacci sums?
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 2002-10-24, 21:51 #1 TTn   22×2,399 Posts Fibonacci sums? Is 2^p -1 always the sum of p Fibonacci numbers? examples: 3=1+2 7=1+1+5 31=2+3+5+8+13 127=1+1+2+13+21+34+55 2047= ...............................? 8191=1+1+5+21+34+55+89+233+377+610+987+1597+4181 I cant seem to find the sum for p=11.
2002-10-25, 03:27   #2
binarydigits

Aug 2002

1101002 Posts
Re: Fibonacci sums?

Quote:
 Originally Posted by TTn Is 2^p -1 always the sum of p Fibonacci numbers?
Yes.

Quote:
 Originally Posted by TTn I cant seem to find the sum for p=11.
There are more.

2002-10-25, 21:47   #3
Maybeso

Aug 2002
Portland, OR USA

2×137 Posts
Re: Fibonacci sums?

Quote:
 Originally Posted by TTn Is 2^p -1 always the sum of p Fibonacci numbers?
Okay, lets make it more interesting:

Is 2^p - 1 always the sum of p Unique* Fibonacci numbers?
*(unique as in use each number once)

The smaller p will be difficult:
7 = 1+1+5 = 2+2+3 ... I see no solution for 3.

To prove or disprove either of these questions, it is sufficient to find the fewest q &lt; p Fibonacci numbers needed to sum each Mp.
i.e. if you can always express Mp as the sum of 5 Fn, then you can replace F(n) with F(n-1) + F(n-2), then repeat the process until you have p numbers.

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