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The purpose of this paper is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation x 2 + d y 6 = z p for square-free values 1 ≤ d ≤ 2 0. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, ℚ -curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations. [ABSTRACT FROM AUTHOR]

Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]