By the trial and error method I have observed the following identity by taking some numerical values. Those are

- $F_m$|$L_n$ is valid only if one of the following holds.

a) $m = 1$ or $m =2$

b) $m = 3$ or $3|n$

c) $n$ is congruent to $2\pmod 4$ and $m = 4$.

The other identity is:

- a) $F_{m+n}$ = $F_{m-1}$ $F_n$ + $F_m$ $F_{n+1}$

b) Either ($L_m$, $F_m$) = 1 or 2.

Unfortunately, I could not get any proofs for the above stated identities. But, numerically and by trial and error methods, the above stated identities are very correct. I am looking for a proof(s) of the above identities...

**edited** The sum of any ten consecutive Fibonacci numbers is always evenly divisible by 11.