Abstract
In this paper, we study some ternary Diophantine equations of the form $$Aa^n+Bb^n=Cc^m$$ , where $$m \in \{2,3\} $$ , and $$n\geqslant 7 $$ is a prime. In fact, we completely solve some particular cases: $$A=5^{\alpha }, ~B=64,~ C=3$$ , when $$m=2$$ ; $$A=2^{\alpha },~ B=27, ~C \in \{7,13\}$$ , when $$m=3$$ .
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