Morley's other miracle: $\displaystyle 4^{p-1}\equiv\pm (\smallmatrix p-1 \frac{p-1}{2} \endsmallmatrix) \pmod {p^3} $

Abstract

Frank Morley is famous for his theorem concerning the angle trisectors of a triangle. This note gives an elementary proof of another result of Morley's, which relates the middle binomial coefficient to a certain power of two. The striking thing about Morley's congruence is that it is valid modulo the third power of the prime being considered.