Monatshefte für Mathematik

, Volume 178, Issue 3, pp 345–359 | Cite as

A von Staudt-type result for \({\sum _{z\in \mathbb {Z}_n[i]} z^k }\)

  • Pedro Fortuny Ayuso
  • José María Grau
  • Antonio M. Oller-Marcén


In this paper we study the sum of powers of 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 result to study the set of integers \(n\) which divide \(\mathbf {G}_n(n)\) and compute its asymptotic density with six exact digits: \(0.971000\ldots \).


Power sum Erdös–Moser equation Asymptotic density 

Mathematics Subject Classification

11B99 11A99 11A07 



The authors wish to thank Jonathan Sondow for his useful comments and remarks.


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Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  • Pedro Fortuny Ayuso
    • 1
  • José María Grau
    • 1
  • Antonio M. Oller-Marcén
    • 2
  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Centro Universitario de la Defensa de ZaragozaZaragozaSpain

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