In a paper of Cohn (see [here](http://www.fq.math.ca/Scanned/2-2/cohn2.pdf)), he uses some formulae involving congruences of Lucas- and Fibonacci-numbers (equations 11,12,13 in the preliminaries section). Does anyone know a source for these (and maybe even for more equations of this type)?



Source for equations involving congruences of Fibonacci and Lucas numbers
(13) follows easily by induction, namely it is true for $m=0$ and $m=1$ by inspection, and then the recursion yields it for all $m$. (11) and (12) hold more generally for all $k$ (including the odd ones) if the minus signs in them are replaced by $(-1)^{k-1}$. Indeed, it is easy to show by a double induction that
$$ F\_{m+2k}+(-1)^k F\_m=L\_k F\_{m+k}\qquad\text{and}\qquad L\_{m+2k}+(-1)^k L\_m=L\_k L\_{m+k},$$
and these imply
$$ F\_{m+2k}\equiv (-1)^{k-1} F\_m\pmod{L\_k}\qquad\text{and}\qquad L\_{m+2k}\equiv (-1)^{k-1} L\_m\pmod{L\_k}.$$
Note that in Cohn's paper $k$ is an even integer.


