# Source for equations involving congruences of Fibonacci and Lucas numbers

In a paper of Cohn (see here), 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)?

(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.