Wolstenholme's theorem over Gaussian integers
Nikita Kalinin
TL;DR
This work introduces a Gaussian-integer analogue of Wolstenholme's theorem by defining $S_p = \sum_{1\le n,m\le p-1,\,(p,m^2+n^2)=1}\frac{1}{n+mi}$ and proving $S_p \equiv 0 \pmod{p^4}$ for primes $p>5$. The authors extend the framework to higher-power sums $S_p^{(k)} = \sum 1/(n+mi)^k$, establishing base-case divisibilities and proposing a general modulus pattern depending on $k\bmod 4$, supported by computational evidence for small primes. They develop auxiliary tools, including eight-term conjugate tuples $T(m,n,k)$ and polynomial-analytic perspectives via $g_p(x)$, to connect these congruences with Gaussian binomial coefficients and potential Bernoulli-number connections. The paper also provides extensive discussions, conjectures, and Macaulay2 code to validate the claimed congruences and to explore extensions to general $n+ni$ and to Gaussian-binomial structures, highlighting potential Lucas-type phenomena in this setting. The results deepen Wolstenholme-type phenomena in the Gaussian integers and suggest rich arithmetic-analytic structures bridging combinatorics, $p$-adic theory, and algebraic number theory.
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.