1
$\begingroup$

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)?

$\endgroup$

1 Answer 1

4
$\begingroup$

(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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.