Skip to main content Accessibility help
×
Home

CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS

  • PAUL M. NANNINGA (a1)

Abstract

In 1903 Mirimanoff conjectured that Cauchy–Mirimanoff polynomials En are irreducible over ℚ for odd prime n. Polynomials Rn, Sn, Tn are introduced, closely related to En. It is proved that Rm, Sm, Tm are irreducible over ℚ for odd m≥3 , and En, Rn, Sn are irreducible over ℚ, for n=2qm, q=1,2,3,4,5 , and m≥1 odd.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS
      Available formats
      ×

Copyright

References

Hide All
[1]Beukers, F., ‘On a sequence of polynomials’, J. Pure Appl. Algebra 117/118 (1997), 97103.
[2]Cauchy, A. and Liouville, J., ‘Rapport sur un mémoire de M. Lamé relatif au dernier théorème de Fermat’, C. R. Acad. Sci. Paris 9 (1839), 359363.
[3]Helou, C., ‘Cauchy–Mirimanoff polynomials’, C. R. Math. Rep. Acad. Sci. Canada 19(2) (1997), 5157.
[4]Irick, B. C., ‘On the irreducibility of the Cauchy–Mirimanoff polynomials’, PhD dissertation, University of Tennessee, Knoxville, 2010.
[5]Klösgen, W., ‘Untersuchungen über Fermatsche Kongruenzen’, Gesellschaft Math. Datenverarbeitung, Nr. 36, Bonn, 1970.
[6]Mirimanoff, D., ‘Sur l’équation (x+1)lx l−1=0’, Nouv. Ann. Math. 3 (1903), 385397.
[7]Ribenboim, P., Fermat’s Last Theorem for Amateurs (Springer, New York, 1999).
[8]Stewart, I. and Tall, D., Algebraic Number Theory and Fermat’s Last Theorem, 3rd edn (A. K. Peters, Natick, MA, 2002).
[9]Terjanian, G., ‘Sur la loi de réciprocité des puissances l-èmes’, Acta Arith. 54(2) (1989), 87125.
[10]Tzermias, P., ‘On Cauchy–Liouville–Mirimanoff polynomials’, Canad. Math. Bull. 50(2) (2007), 313320.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS

  • PAUL M. NANNINGA (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed