FT18: Divisibility Properties for the Lucas Numbers

Two other formulas from part 16 contain terms with the factor F(m), and so can be treated similarly to the previous post. These are:

• F(kn + m) ≡ (−1)^(n+1)k/2 F (m) (mod L(n)),
  for k even.

• L(kn+m) ≡ (−1)^{(n+1)(k−1)/2} F(m) L(n−1) (mod L(n)),
  for k odd.

Take m = 0 and use the fact that F(0) = 0 to see that:

• L(n) | F (kn) for k even.

• L(n) | L (kn) for k odd.