Monatshefte für Mathematik

, Volume 177, Issue 3, pp 421–436 | Cite as

On the congruence \(1^m + 2^m + \cdots + m^m\equiv n \pmod {m}\) with \(n\mid m\)

  • José María Grau
  • Antonio M. Oller-Marcén
  • Jonathan Sondow


We show that if the congruence above holds and \(n\mid m\), then the quotient \(Q:=m/n\) satisfies \(\sum _{p\mid Q} \frac{Q}{p}+1 \equiv 0\pmod {Q}\), where \(p\) is prime. The only known solutions of the latter congruence are \(Q=1\) and the eight known primary pseudoperfect numbers 2, 6, 42, 1806, 47058, 2214502422, 52495396602, and 8490421583559688410706771261086. Fixing \(Q\), we prove that the set of positive integers \(n\) satisfying the congruence in the title, with \(m=Q n\), is empty in case \(Q=\) 52495396602, and in the other eight cases has an asymptotic density between bounds in \((0,1)\) that we provide.


Power sum Congruence Erdős–Moser equation  Asymptotic density 

Mathematics Subject Classification (2010)

11B99 11A99 11A07 



We are grateful to the anonymous referee for suggestions on asymptotic density and, especially, for the proof of Theorem 3. We thank Bernd Kellner for useful comments on several references.


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Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • José María Grau
    • 1
  • Antonio M. Oller-Marcén
    • 2
  • Jonathan Sondow
    • 3
  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Centro Universitario de la Defensa de ZaragozaZaragozaSpain
  3. 3.New YorkUSA

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