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.

[binarydigits]

Originally Posted by TTn

Is 2^p -1 always the sum of p Fibonacci numbers?

Yes.

Originally Posted by TTn

I cant seem to find the sum for p=11.

One answer is 2047=2+3+5+8+21+34+55+89+233+610+987.
There are more.

[Maybeso]

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 < 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.