Combinatorial Proofs of Fermat's, Lucas's, and Wilson's Theorems

The Lucas numbers, 2, 1, 3, 4, 7, 11, 18, 29, 47, . . . , named in honor of Edouard Lucas (1842-1891), are defined by L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. It is easy to show that, for n ≥ 1, Ln counts the ways to create a bracelet of length n using beads of length one or two, where bracelets that differ by a rotation or a reflection are still considered distinct. For example, there are four bracelets of length three. (Such a bracelet can have three beads of length one, or it can have a bead of length two and a bead of length one, where the bead of length one can be in position one, two, or three.) Let f act on bracelets of prime length p by rotating each bead clockwise one unit. Clearly f leaves any bracelet unchanged. Since f has just one fixed point (when all beads have length one), we conclude that Lp ≡ 1 (mod p) for each prime p. More generally, as defined in [4], for nonnegative integers a and b, the Lucas sequence (of the second kind) is defined by V0 = 2, V1 = a, and Vn = aVn−1 + bVn−2 for n ≥ 2. Again, it is easy to show [1] that Vn with n ≥ 1 counts colored bracelets of length n, where there are a color choices for beads of length one and b color choices for beads of length two. By the same argument as earlier, with the exception of those bracelets consisting of length one beads all of the same color, when p is prime every bracelet can be rotated to create p distinct bracelets. Thus, for p prime,