On the exponential Diophantine equation ${\left(\frac{q^{2l}-p^{2k}}{2}n\right)}^x+(p^k{q}^{l}n{)}^y={\left(\frac{q^{2l}+p^{2k}}{2}n\right)}^z$

Let $k,l,m_1,m_2$ be positive integers and let both $p$ and $q$ be odd primes such that $p^k=2^{m_1}-a^{m_2}$ and $q^l=2^{m_1}+a^{m_2}$ where $a$ is odd prime with $a\equiv 5(\mathrm{mod}8)$ and $a\not\equiv 1(\mathrm{mod}5)$. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that the exponential Diophantine equation ${\left(\frac{q^{2l}-p^{2k}}{2}n\right)}^x+(p^k{q}^{l}n{)}^y={\left(\frac{q^{2l}+p^{2k}}{2}n\right)}^z$ has only the positive integer solution $(x,y,z)=(2,2,2)$.