LAUSR

Abstract We consider the Diophantine equation 7 x 2 + y 2 n = 4 z 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 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper integer solutions for specific primes p > 7 . We computationally verify the criterion for all primes 7 p 10 9 , p ≠ 13 . We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7 x 2 + y 2 p = 4 z 3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x 2 + 7 y 2 n = 4 z 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 .