Congruence Properties of Fibonacci Numbers and Fibonacci Coefficients Modulo p²

Abstract

Among the many fascinating properties of the binomial coefficients, one of the most striking is that

$$ \left( {\begin{array}{*{20}{c}} {ap} \\ {bp} \\ \end{array} } \right) \equiv \left( {\begin{array}{*{20}{c}} a \\ b \\ \end{array} } \right)\;\left( {\bmod \;{p^k}} \right) $$

(1)

for k = 1, k = 2 and if p > 3 for k = 3, [1] [2] [5] [10].