{"paper":{"title":"A von Staudt-type formula for $\\displaystyle{\\sum_{z\\in\\mathbb{Z}_n[i]} z^k }$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonio Oller-Marcen, Jose Maria Grau, Pedro Fortuny Ayuso","submitted_at":"2014-02-03T10:27:19Z","abstract_excerpt":"In this paper we study the sum of powers in the Gaussian integers $\\mathbf{G}_k(n):=\\sum_{a,b \\in [1,n]} (a+b i)^k$. We give an explicit formula for $\\mathbf{G}_k(n) \\pmod n $ in terms of the prime numbers $p \\equiv 3 \\pmod 4$ with $p \\mid \\mid n$ and $p-1 \\mid k$, similar to the well known one due to von Staudt for $\\sum_{i=1}^n i^k \\pmod n$. We apply this formula to study the set of integers $n$ which divide $\\mathbf{G}_n(n)$ and compute its asymptotic density with six exact digits: $0.971000\\ldots$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.0333","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}