Wolstenholme's theorem over Gaussian integers

2025/03/30 by Kalinin, Nikita · 1 citation
#11A07 11D79 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.07978

Abstract

This paper establishes an extension of Wolstenholme's theorem to the ring of Gaussian integers ℤ[i]. For a prime p > 7, we prove that the sum S_p of inverses of Gaussian integers in the set \n+mi | 1 ≤ n, m ≤ p-1, gcd(p, mi+n)=1\ satisfies the congruence S_p ≡ 0 \pmodp⁴. 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 ≤ 1000) are explicitly computed and tabulated.

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