Wolstenholme's theorem over Gaussian integers

Abstract

This paper establishes an extension of Wolstenholme's theorem to the ring of Gaussian integers $\mathbb{Z}[i]$. For a prime $p > 7$, we prove that the sum $S_p$ of inverses of Gaussian integers in the set $\{n+mi \mid 1 \leq n, m \leq p-1, \gcd(p, mi+n)=1\}$ satisfies the congruence $S_p \equiv 0 \pmod{p^4}$. We further generalize this result to higher-power sums $S_p^{(k)}$, demonstrating structured divisibility patterns modulo powers of $p$. We propose some conjectures generalising the connections between classical Wolstenholme's theorem and binomial coefficients. Special cases and irregularities for small primes ($p \leq 1000$) are explicitly computed and tabulated.