 
Summary: COMBINATORIAL PROOFS OF FERMAT'S, LUCAS'S, AND WILSON'S
THEOREMS
PETER G. ANDERSON, ARTHUR T. BENJAMIN, AND JEREMY A. ROUSE
In this note, we observe that many classical theorems from number theory are simple consequences
of the following combinatorial lemma.
Lemma 1. Let S be a finite set, let p be prime, and suppose f : S S has the property that
fp
(x) = x for any x in S, where fp
is the pfold composition of f. Then S F (mod p), where
F is the set of fixed points of f.
Proof. S is the disjoint union of sets of the form {x, f(x), . . . , fp1
(x)}. Since p is prime, each set
either has size one or size p.
The Lucas numbers, 2, 1, 3, 4, 7, 11, 18, 29, 47, . . . , named in honor of Edouard Lucas (18421891),
are defined by L0 = 2, L1 = 1, and Ln = Ln1 + Ln2 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.
