Abstract
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.
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Notes
As a confirmation of our results, we verified our results with Magma for \(7\le n\le 1000\) and found identical results.
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Acknowledgements
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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Kapetanakis, G. A Swan-like note for a family of binary pentanomials. AAECC 30, 361–372 (2019). https://doi.org/10.1007/s00200-018-0378-7
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DOI: https://doi.org/10.1007/s00200-018-0378-7