Preda Milhaescu is a good example http://en.wikipedia.org/wiki/Preda_Mihăilescu.http://en.wikipedia.org/wiki/Preda_Mihăilescu.
He received his PhD with 42, and proved the Catalan conjecture 5 years later.
The Catalan conjecture was open for 160 years.
He proposed in 2009 a proof of the Leopoldt conjecture, but I am not sure about the status of this.

