EXISTENCE OF EVEN PERFECT POLYNOMIALS

Luis H. Gallardo, Olivier Rahavandrainy, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

Perfect polynomials are a natural analogue (in the ring Fp[x]) of multiperfect numbers (in the ring of integers). The latter numbers are classical objects that are poorly understood, since only their definition is simple. We describe, by elementary methods, the most basic objects in the polynomial case of the general problem. We display, for every prime number p ≢ 1 mod 12 (resp. p ≢ 1 mod 24) many new even non-splitting perfect (resp. unitary perfect) polynomials over Fp. Moreover, for any prime number p ≢ 1 mod 24, new bi-unitary perfect polynomials are also given. These examples substantially improve our knowledge about these kinds of polynomials.

Original languageEnglish (US)
Pages (from-to)47-62
Number of pages16
JournalMathematical Reports
Volume25
Issue number1
DOIs
StatePublished - 2023

Keywords

  • (bi-unitary) perfect polynomials
  • characteristic p
  • finite fields

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Applied Mathematics

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