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 non-trivial proper integer solutions for specific primes $p>7$. We computationally verify the criterion for all primes $7<p<{10}^9$, $p\ne 13$. We use the symplectic method and quadratic reciprocity to show that the Diophantine equation $7x^2+y^{2p}=4z^3$ has no non-trivial proper solutions for a positive proportion of primes p. In the paper [10] we consider the Diophantine equation $x^2+7y^{2n}=4z^3$, 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 $n\ge 5$.

我们考虑丢番图方程$7X^2+{\mathrm{是的}}^{2n}=4z^3$. 我们确定这个方程的所有解$n=2,3,4$和 5. 我们制定了一个克劳斯类型标准来证明丢番图方程$7X^2+{\mathrm{是的}}^{2p}=4z^3$对特定素数没有非平凡的适当整数解$p>7$. 我们通过计算验证所有素数的标准$7<p<{10}^9$,$p\ne 13$. 我们使用辛方法和二次互易性来证明丢番图方程$7X^2+{\mathrm{是的}}^{2p}=4z^3$对于正比例的素数p没有非平凡的适当解. 在论文 [10] 中，我们考虑丢番图方程$X^2+7{\mathrm{是的}}^{2n}=4z^3$，确定所有解决方案族 $n=2$ 和 3，以及给出解决方案的（大部分）猜想描述 $n=4$ 和素数 $n\ge 5$.