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-   -   Fibonacci sums? (https://www.mersenneforum.org/showthread.php?t=178)

TTn 2002-10-24 21:51

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. :rolleyes:

binarydigits 2002-10-25 03:27

Re: Fibonacci sums?
 
[quote="TTn"]Is 2^p -1 always the sum of p Fibonacci numbers?[/quote]
Yes.

[quote="TTn"]I cant seem to find the sum for p=11.[/quote]
One answer is 2047=2+3+5+8+21+34+55+89+233+610+987.
There are more.

Maybeso 2002-10-25 21:47

Re: Fibonacci sums?
 
[quote="TTn"]Is 2^p -1 always the sum of p Fibonacci numbers?[/quote]
Okay, lets make it more interesting:

Is 2^p - 1 always the sum of p [b]Unique* [/b]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 [b]fewest [/b]q < 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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