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

JOURNAL OF NUMBER THEORY

卷:234

Article

Chalupka, Karolina¹  Dabrowski, Andrzej¹  Soydan, Gokhan² 

[1] Univ Szczecin, Inst Math, Wielkopolska 15, PL-70451 Szczecin, Poland

[2] Bursa Uludag Univ, Dept Math, TR-16059 Bursa, Turkey

关键词: Diophantine equation;    Modular form;    Elliptic curve;    Galois representation;    Chabauty method;   

DOI  :  10.1016/j.jnt.2021.06.019

来源: Elsevier

【 摘 要 】

We 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.(C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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