7
$\begingroup$

I have the following conjecture, need to know a proof just in case mine is wrong, or the conjecture itself is wrong.

The sum of $k$ distinct Fibonacci numbers can be written in at most $k$ ways as the sum of another $k$ distinct numbers from the sequence

Proof: Take, say 4, Fibonacci numbers with distinct values: $$144+34+3+2$$ We can see that we must maintain the distinctness and number of the Fibonacci numbers. So when we split the numbers; $$[89+55]+[13+21]+3+2$$

It is immediately apparent that if some numbers are split into smaller Fibonacci numbers, others must be merged to keep only 4 numbers. Thus, 3+2 becomes 5, and either 144 or 34 kept as-is. Thus, $$144+34+3+2=89+55+34+5=144+21+13+5$$

I know it's a rudimentary proof, and $k$ ways of writing the same sum is probably a wrong upper bound, but at least it works for now.

Please do let me know what's wrong. Thanks.

  • 1
    Use single dollar signs, e.g. `$k$`, for **inline** mathematics, and use double dollar signs, e.g. `$$a^2+b^2=c^2$$`, for mathematics you want to display on a separate line.2012-07-08

2 Answers 2