Catalan's conjecture asserts that $3^2-2^3=1$ is the only solution to the equation $x^m-y^n=1$ for natural numbers $x$, $y$, $m$, $n$, where $m, n > 1$. The conjecture was proven by Preda Mihailescu in 2002. In the thesis, we present the history of solving Catalan's conjecture and examine individual key results. We prove Catalan's conjecture in cases where one of the exponents is an even number. Further resolution of Catalan's conjecture relies on results from algebraic number theory, particularly the theory of cyclotomic fields. Among other things, we prove Cassels' divisibility theorem, which states that if nonzero integers $x$ and $y$ are solutions to the equation $x^p-y^q=1$, where $p$ and $q$ are odd primes, then $p$ divides $y$ and $q$ divides $x$. We also prove Mihailescu's divisibility theorem for the same equation, stating that $q^2$ divides $x$ and $p^2$ divides $y$. Mihailescu's final proof of Catalan's conjecture consists of two parts. In the master's thesis, we fully present the part where $p$ divides $q-1$ and a part of the proof of the second part when $p$ does not divide $q-1$, using Runge's method.
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