On some ternary Diophantine equations of Signature $(p,p,k)$

In this paper, we summarize the work on ternary Diophantine equation of the form $A{x}^n+B{y}^n=c{z}^m$, where $m\in \{2,3,n\}$, $n⩾7$ is a prime. Moreover, we completely solve some particular cases ($A=5^{\alpha },B=64,c=3,m=2;A=2^{\alpha },B=27,c\in \{7,13\},m=3$).