Cubic forms, powers of primes and classification of elliptic curves
Tom 166 / 2021
Colloquium Mathematicum 166 (2021), 137-150
MSC: Primary 11D41.
DOI: 10.4064/cm8132-8-2020
Opublikowany online: 11 March 2021
Streszczenie
We prove that for almost all primes the equation (x+y)(x^2+Bxy+y^2)=p^\alpha z^n with B\in \{0,1,4,6\} has no solution in pairwise coprime nonzero integers x, y, z\neq \pm 1, integer \alpha \gt 0 and prime n\geq p^{16p}. Proving this we classify primes p for which there exists an elliptic curve over \mathbb {Q} with conductor 48p, 192p or 384p and with nontrivial rational 2-torsion.
