# Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a Greek letter that has something to do with what the function is named after, I was thinking $\lambda(m)$.

• ...not to derail your work too much, but won't this be exactly the same as the Pisano Period? Since the Lucas numbers can be written as a linear combination of the Fibonaccis and vice versa, any period of one is a period of the other... Jan 14, 2014 at 23:17
• Haha, if that's true, then you'd be saving me more work, not derailing my current work. Unfortunately, it's not true. $\pi(10) = 60$ while $\lambda(10) = 12$. Jan 14, 2014 at 23:18
• Well, since $L_n=F_{n-1}+F_{n+1}$, if we have $F_k\equiv F_{k+p}$ then $L_{k+p}=F_{k+p-1}+F_{k+p+1}$ $\equiv F_{k-1}+F_{k+1}$ $\equiv L_k$. A similar formula will go the other way, but since the Lucas-to-Fibonacci conversion involves a division by 5, it might be a little bit trickier if $m$ is divisible by 5... Jan 14, 2014 at 23:21
• Whoops. :-) Though it is in fact correct that $\lambda(m)|\pi(m)$ for all $m$ by the Fibonacci-to-Lucas formula, and that they'll be equal for all $m$ with $(m,5)=1$... Jan 14, 2014 at 23:22
• What do you mean $(m,5) = 1$? Jan 14, 2014 at 23:29