Lucas' Theorem modulo $p^n$ via Harmonic sums

Lucas’ theorem states that for a prime $p$, integers $a,b\ge 0$ and $0\le r,s\le p-1$, one has $\left(\genfrac{}{}{0px}{}{pa+r}{pb+s}\right)\equiv \left(\genfrac{}{}{0px}{}{a}{b}\right)\left(\genfrac{}{}{0px}{}{r}{s}\right)p$. Let $D(p^n)$ be the set of pairs $(r,s)$ with $s\le r$ for which this congruence holds modulo $p^n$ for all $a,b\ge 0$. Rowland characterized $D(p^2)$ in terms of harmonic sums and symmetries of Pascal’s triangle. For $n<p$, we give a criterion for a point $(r,s)$ to lie in $D(p^n)$ in terms of generalized harmonic sums of order at most $n-1$. We show that $D(p^n)$ is invariant under the reflection $(r,s)\mapsto (r,r-s)$ and the rotation $(r,s)\mapsto (p-1-s,r-s)$, so that with every point it contains its full orbit under the dihedral group $D_3$. Applied to the vertices, our criterion gives a result of Jacobsthal-Ljunggren type modulo $p^n$. In particular, we recover Zhao’s result $(0,0)\in D(p^4)$ if and only if $p$ is a Wolstenholme prime. We also study the centroid and the edge midpoints, whose membership in $D(p^3)$ and $D(p^4)$ is governed by Bernoulli numbers and polynomials, Wolstenholme primes and super Wieferich primes. In particular, for $p=205129$ the centroid lies in $D(p^3)$, answering a question of Rowland.