On a class of generalized Fermat equations of signature (2,2n,3)

• Published in: Journal of number theory Vol. 234; pp. 153 - 178
• Main Authors: Chałupka, Karolina, Dąbrowski, Andrzej, Soydan, Gökhan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2022
• Subjects: Chabauty method; Diophantine equation; Elliptic curve; Galois representation; Modular form; 11B39; Chabauty method; 11D61; Galois representation; Diophantine equation; Elliptic curve; Modular form
• ISSN: 0022-314X; 1096-1658
• DOI: 10.1016/j.jnt.2021.06.019

We consider the Diophantine equation 7x2+y2n=4z3. 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 7x2+y2p=4z3 has no non-trivial proper integer solutions for specific primes p>7. We computationally verify the criterion for all primes 7<p<109, p≠13. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation 7x2+y2p=4z3 has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation x2+7y2n=4z3, 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.