On the conjecture of Je $\acute{s}$manowicz

Let $k,l,m_1$ and $m_2$ 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 a positive integer with $a\equiv \pm 3(\mathrm{mod}8)$. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je $\acute{s}$manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:

$$\frac{q^{2l}-p^{2k}}{2},p^k{q}^l,\frac{q^{2l}+p^{2k}}{2}.$$

We also prove that Je $\acute{s}$manowicz' conjecture holds for non-primitive Pythagorean numbers:

$$n\frac{q^{2l}-p^{2k}}{2},n{p}^k{q}^l,n\frac{q^{2l}+p^{2k}}{2},$$

for any positive integer $n$ if for $a=a_1{a}_2$ with $a_1\equiv 1(\mathrm{mod}8)$ not a square and $gcd(a_1,a_2)=1$, then there exists a prime divisor $P$ of $a_2$ such that $\left(\frac{a_1}{P}\right)=-1$ and $2|m_1,a\equiv 5(\mathrm{mod}8)$ or $2\text{⧸}|m_2,a\equiv 3(\mathrm{mod}8)$.