Chalupka, KarolinaDabrowski, Andrzej2024-09-302024-09-302022-01-250022-314Xhttps://doi.org/10.1016/j.jnt.2021.06.019https://hdl.handle.net/11452/45454We consider the Diophantine equation 7x(2) + y(2n) = 4z(3). We determine all solutions to this equation for n = 2, 3, 4 and 5. We formulate a Kraus type criterion for showing that the Diophantine equation 7x(2) + y(2p) = 4z(3) has no nontrivial proper integer solutions for specific primes p > 7. We computationally verify the criterion for all primes 7 < p < 10(9), p &NOTEQUexpressionL;13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7x(2) + y2p = 4z3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x(2) + 7y(2n) = 4z(3), determining all families of solutions for n = 2 and 3, as well as giving a (mostly) conjectural description of the solutions for n = 4 and primes n >= 5.eninfo:eu-repo/semantics/closedAccessDiophantine equationsDiophantine equationModular formElliptic curveGalois representationChabauty methodScience & technologyPhysical sciencesMathematicsArticle00079590830000815317823410.1016/j.jnt.2021.06.019