An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.
Two notable examples in mathematics that have been solved and "closed" by researchers in the late twentieth century are Fermat's Last Theorem and the four color map theorem.
He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve.
Attempts to prove it prompted substantial development in number theory, and over time Fermat's Last Theorem itself gained legendary prominence as an unsolved problem in popular mathematics.