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 $\{n+mi\mid 1\le n,m\le p-1,\gcd (p,mi+n)=1\}$ satisfies the congruence $S_p\equiv 0(\mathrm{mod}{p}^4)$. 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 ($p\le 1000$) are explicitly computed and tabulated.