Fermat's not-so-little theorem

[Larry Hammick]

I haven't got around to TeXing out a proof of this thing, but I'm eager to
show it off anyhow.

Theorem: Let f(A,B,...) be a polynomial in any number of variables over Z,
and let
n be a positive integer. Define a polynomial F by

F(A,B,...) = sum_{d|n} M(d) f(A^d, B^d, ...)^{n/d}

where M denotes the Mobius function. Then all the coefficients of F are
divisible by n.

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Fermat's little theorem is the special case in which n is a prime and f is a
constant! :D

-- Larry