Sunday, August 8, 2010

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.

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