In this talk, we give a survey about pure ternary exponential Diophantine equations and some developments in results as well as in tools. These tools include Baker's method, the Bilu-Hanrot-Voutier Theorem about the existence of primitive divisors of Lucas numbers and Lehmer numbers, algebraic number theory methods, and others. We generalize the conjectures of Jesmanowicz, Terai-Cao-Le and Yuan-Han and also present some results that we proved and plan to further improve in this area.
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