Abstract
Let p be a prime number. In the early 2000s, it was proved that the Fermat equations with coefficients3xp+8yp+21zp=0 and 3xp+4yp+5zp=0 do not admit non-trivial solutions for a set of exponents p with Dirichlet density 1/4 and 1/8, respectively. In this note, using a recent criterion to decide if two elliptic curves over Q with certain types of additive reduction at 2 have symplectically isomorphic p-torsion modules, we improve these densities to 3/8.
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