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

Abstract

Given integers ℓ > m > 0, monic polynomials Xn, Yn, and Zn are given with the property that the complex number ρ is a zero of Xn if and only if the triple (ρ, ρ + m, ρ + ℓ) 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.

Keywords

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

2010 Mathematics Subject Classification

  • 11A99
  • 11C08
  • 11D41

References

  1. Cox, D. A. (2011). Why Eisenstein proved the Eisenstein criterion and why Schonemann discovered it first [reprint of mr2572615]. Amer. Math. Monthly, 118(1), 3–21.
  2. Golomb, S. W. (1970). Powerful numbers. Amer. Math. Monthly, 77, 848–855.
  3. Paparella, P. Is every powerful number the sum of a powerful number and a prime? MathOverflow. Available online: https://mathoverflow.net/q/269080 (version: 2017-05-07).
  4. Prasolov, V. V. (2010). Polynomials Algorithms and Computation in Mathematics, 11, Springer-Verlag, Berlin. Translated from the 2001 Russian second edition by Dimitry Leites, Paperback edition [of MR2082772].
  5. Ribenboim, P. (1999). Fermat’s last theorem for amateurs. Springer-Verlag, New York.
  6. Taylor, R. & Wiles, A. (1995). Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2), 141(3), 553–572.
  7. Wiles, A. (1995). Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3), 443–551.

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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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