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

TL;DR

This paper presents an elementary proof of Morley's congruence, a remarkable number-theoretic result relating binomial coefficients to powers of two modulo the cube of a prime, extending Morley's mathematical legacy.

Contribution

It provides a new elementary proof of Morley's congruence involving binomial coefficients and prime powers, which was previously established through more complex methods.

Findings

Proves Morley's congruence modulo p^3

Shows the relation between binomial coefficients and powers of two

Extends understanding of prime power congruences

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.