In 1844, Belgium mathematician Eug`ene Charles Catalan submitted a conjecture to Crelle’s Journal. The conjecture states “Two consecutive integers, other than 8 and 9, cannot be exact powers. In other words, the equation xm-yn =1 admits only one solution in the positive integers.” In 1850, V. A. Lebesgue proved the case for xm-y2 =1, and in 1965, Ko Chao showed that x2-yn = 1 has no solution in positive integers x, y, and n with n > 1. Finally, in 2002, Preda V. Mihailescu proved the conjecture.
In this thesis, we explore the proof of this famous statement.

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