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Never Underestimate A Theorem That Counts Something!, Tyler J. Evans
Never Underestimate A Theorem That Counts Something!, Tyler J. Evans
Tyler Evans
In their (wonderful) note Combinatorial proofs of Fermat’s, Lucas’s and
Wilson’s theorems (MONTHLY, March 2005), Anderson, Benjamin and
Rouse derive a host of classical divisibility theorems all from the following
result:
lemma. If X is a finite set, p a prime integer and f : X ! X a mapping
satisfying fp(x) = x for all x 2 X, then |X| #17; |X0| (mod p), where
X0 = {x 2 X | f(x) = x} denotes the set of fixed points of f.
Remark. |X| #17; |X0| (mod p) () |X| + (p − 1)|X0| #17; 0 (mod p).
ABR: …
Binomial Identities With Pascalgt, Tyler J. Evans