Eisenstein’s criterion, Fermat’s last theorem, and a conjecture on powerful numbers

Pietro Paparella
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 22—29
DOI: 10.7546/nntdm.2019.25.2.22-29
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Authors and affiliations

Pietro Paparella 
Division of Engineering and Mathematics, University of Washington Bothell
18115 Campus Way NE, Bothell, WA 98011-8246, United States


Given integers  > > 0, monic polynomials XnYn, and Zn are given with the property that the complex number ρ is a zero of Xn if and only if the triple (ρρ + mρ + ) satisfies xn + yn = zn. It is shown that the irreducibility of these polynomials implies Fermat’s last theorem. It is also demonstrated, in a precise asymptotic sense, that for a majority of cases, these polynomials are irreducible via application of Eisenstein’s criterion. We conclude by offering a conjecture on powerful numbers.


  • Eisenstein’s criterion
  • Fermat’s last theorem
  • Fermat equation
  • Irreducible polynomial
  • Powerful numbers

2010 Mathematics Subject Classification

  • 11A99
  • 11C08
  • 11D41


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Cite this paper

Paparella, P. (2019). Eisenstein’s criterion, Fermat’s last theorem, and a conjecture on powerful numbers. Notes on Number Theory and Discrete Mathematics, 25(2), 22-29, doi: 10.7546/nntdm.2019.25.2.22-29.

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