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CAUCHY–MIRIMANOFF AND RELATED POLYNOMIALS

Part of: Polynomials and matrices

Published online by Cambridge University Press:  27 September 2012

PAUL M. NANNINGA
PAUL M. NANNINGA*
Affiliation:
Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia (email: paul.nanninga@anu.edu.au)
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Abstract

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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.

Keywords

irreducible polynomialsCauchy–Mirimanoff polynomialFermat’s last theorem

MSC classification

Secondary: 11C08: Polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 269 - 280
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Beukers, F., ‘On a sequence of polynomials’, J. Pure Appl. Algebra 117/118 (1997), 97–103.Google Scholar
[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), 359–363.Google Scholar
[3]Helou, C., ‘Cauchy–Mirimanoff polynomials’, C. R. Math. Rep. Acad. Sci. Canada 19(2) (1997), 51–57.Google Scholar
[4]Irick, B. C., ‘On the irreducibility of the Cauchy–Mirimanoff polynomials’, PhD dissertation, University of Tennessee, Knoxville, 2010.Google Scholar
[5]Klösgen, W., ‘Untersuchungen über Fermatsche Kongruenzen’, Gesellschaft Math. Datenverarbeitung, Nr. 36, Bonn, 1970.Google Scholar
[6]Mirimanoff, D., ‘Sur l’équation (x+1)l−x l−1=0’, Nouv. Ann. Math. 3 (1903), 385–397.Google Scholar
[7]Ribenboim, P., Fermat’s Last Theorem for Amateurs (Springer, New York, 1999).Google Scholar
[8]Stewart, I. and Tall, D., Algebraic Number Theory and Fermat’s Last Theorem, 3rd edn (A. K. Peters, Natick, MA, 2002).Google Scholar
[9]Terjanian, G., ‘Sur la loi de réciprocité des puissances l-èmes’, Acta Arith. 54(2) (1989), 87–125.Google Scholar
[10]Tzermias, P., ‘On Cauchy–Liouville–Mirimanoff polynomials’, Canad. Math. Bull. 50(2) (2007), 313–320.Google Scholar