JournalscmhVol. 91, No. 2pp. 295–304

On the Fermat-type equation x3+y3=zpx^3 + y^3 = z^p

  • Nuno Freitas

    The University of British Columbia, Vancouver, Canada
On the Fermat-type equation $x^3 + y^3 = z^p$ cover

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Abstract

We prove that the Fermat-type equation x3+y3=zpx^3 + y^3 = z^p has no solutions (a,b,c)(a,b,c) satisfying abc0abc \neq 0 and gcd(a,b,c)=1(a,b,c) = 1 when –3 is not a square mod pp. This improves to approximately 0.844 the Dirichlet density of the set of prime exponents to which the previous equation is known to not have such solutions.

For the proof we develop a criterion of independent interest to decide if two elliptic curves with certain type of potentially good reduction at 2 have symplectically or anti-symplectically isomorphic pp-torsion modules.

Cite this article

Nuno Freitas, On the Fermat-type equation x3+y3=zpx^3 + y^3 = z^p. Comment. Math. Helv. 91 (2016), no. 2, pp. 295–304

DOI 10.4171/CMH/386