Frits Beukers

We discuss generalizations of the well-known congruences

u(mp^r) = u(mp^(r-1)) mod p^r of the Fibonacci sequence u(1)=1, u(2)=3,

u(3)=4, ... Here m,r are arbitrary positive integers and p is an arbitrary prime.

The generating function of the u(n) is (x+2x^2)/(1-x-x^2). It turns out

that similar congruences occur for coefficients of certain multivariable rational functions.

We explore this phenomenon. The ultimate goal will be to find cases where

congruences mod p^(2r) and p^(3r) hold.