Skip to main content
SpringerLink
Log in
Book cover

Number Theory and Discrete Mathematics pp 179–187Cite as

Birkhäuser

The Lehmer Problem on the Euler Totient: A Pendora’s Box of Unsolvable Problems

  • Conference paper

  • 572 Accesses

Part of the book series: Trends in Mathematics ((TM))

Abstract

The celebrated seventy year old, innocent looking problem of D.H. Lehmer [5] asking for composite numbers, if any, satisfying the relation Ø(n)| (n — 1), where Ø(n) is the Euler totient, is still unsolved. This is easily seen to be equivalent to asking the Problem 1.1. Given n > 1, n odd and Ø(n)| (n — 1), is n necessarily a prime?

Article FootNote

Supported in part by the Natural Sciences and Engineering Research Grant of Canada.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G.L. Cohen and Hagis, Peter, Jr., On the number of prime factors of n with φ(n) | n -1,Nieuw Archief Voor Wiskunde, XXVII(3), 177–185, 1980.

    MathSciNet  Google Scholar 

  2. Hagis, Peter, Jr., On the equation Mφ(n) = n - 1, Nieuw Archief Voor Wiskunde, 6 255–261, 1988.

    MathSciNet  MATH  Google Scholar 

  3. M. Kishore, On the number of distinct prime factors of n for which φ(n)|n - 1, Nieuw Archief Voor Wiskunde, XXV(3), 18–52, 1972.

    Google Scholar 

  4. E. Landau, Elementary Number Theory, (Second Edition), Chelsea.

    Google Scholar 

  5. D.H. Lehmer, On Euler’s totient function, Bull. Amer. Math. Soc. 38 745–751, 1932.

    Article  MathSciNet  Google Scholar 

  6. E. Lieuwens, Do there exist composite M for which kφ(M) = M - 1 holds?, Nieuw Archief Voor Wiskunde, XVIII(3), 165–169, 1970.

    MathSciNet  Google Scholar 

  7. Narkiewicz, On a class of arithmetical convolutions, Coll. Math. 10 81–94, 1963.

    MathSciNet  MATH  Google Scholar 

  8. C. Pomerance, On composite n for which φ(n) |n -1,II Pacific J. Math. 69 177–186, 1977.

    Article  MathSciNet  Google Scholar 

  9. Fr. Schuh, Do there exist composite numbers m for which φ(m)|m - 1?, (Dutch), Mathematica Zutpen, B13 102–107, 1944.

    Google Scholar 

  10. V. Siva Rama Prasad and M. Rangamma, On composite n satisfying a problem of Lehmer, Indian J. Pure. Math. 16(11), 1244–1248, 1985.

    MathSciNet  MATH  Google Scholar 

  11. V. Siva Rama Prasad and M.V. Subbarao, Regular convolutions and a related Lehmer problem, Nieuw Archief Voor Wiskunde, 3(4), 1–18, 1985.

    MathSciNet  MATH  Google Scholar 

  12. V. Siva Rama Prasad and M. Rangamma, On composite n for which φ(n)|n - 1, Nieuw Archief Voor Wiskunde, V(4), 77–81, 1989.

    Google Scholar 

  13. V. Siva Rama Prasad and M. Rangamma, On the forms of n for which φ(n)|n - 1, Indian J. Pure Maths. 20(9), 871–873, 1989.

    MathSciNet  MATH  Google Scholar 

  14. M.V. Subbarao, On the problem concerning unitary totient function φ* (n), Notices Amer. Math. Soc. 18 940, 1971.

    Google Scholar 

  15. M.V. Subbarao and V. Siva Rama Prasad, Some analogues of a Lehmer problem on the totient function, Rocky Mountain Journal Math. 15 609–620, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  16. M.V. Subbarao, A companion to a Lehmer Problem, Publicationes Math. Debrecen, 52, 683–698, 1998.

    MathSciNet  MATH  Google Scholar 

  17. M.V. Subbarao, Are there an infinity of unitary perfect numbers?, Amer Math. Monthly, 77, 389–390, 1970.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Alberta, Edmonton, Alberta, T6G 2G1, Canada

    M. V. Subbarao

Authors
  1. M. V. Subbarao
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, 160014, India

    A. K. Agarwal

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer Basel AG

About this paper

Cite this paper

Subbarao, M.V. (2002). The Lehmer Problem on the Euler Totient: A Pendora’s Box of Unsolvable Problems. In: Agarwal, A.K., Berndt, B.C., Krattenthaler, C.F., Mullen, G.L., Ramachandra, K., Waldschmidt, M. (eds) Number Theory and Discrete Mathematics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8223-1_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8223-1_18

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9481-4

  • Online ISBN: 978-3-0348-8223-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation