A Swan-like note for a family of binary pentanomials

  • Giorgos KapetanakisEmail author
Original Paper


In this note, we employ the techniques of Swan (Pac J Math 12(3):1099–1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial \(X^n+X^{3s}+X^{2s}+X^{s}+1\in \mathbb {F}_2[X]\), where s is even and \(n>3s\). Our results imply that if \(n \not \equiv \pm 1 \pmod {8}\), then the polynomial in question is reducible.


Swan-like Binary field Pentanomial 

Mathematics Subject Classification

11T06 11C08 



This work was initiated during the author’s visit to the Federal University of Santa Catarina. The author is grateful to the anonymous reviewers for their valuable comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering and Natural SciencesSabancı ÜniversitesiTuzlaTurkey

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