Another example is the proof of Fermat's last theorem by Andrew Wiles. Since the proof is long and involved, it took the author himself to point out the errors, and later fix them.
I take it you are advocating that the only good way to ensure a program is correct is to run it on a computer. But here's an inherent problem: your mental model of the semantics of the programming language you use and what is actually implemented on the computer are bound to differ. Which one is correct is subject to debate (i.e. whether the documentation -- which is the only thing consumable by humans -- or the implementation -- which is consumable by computers -- is definitive). A document may be ambiguous, yet a human reading it can still understand what is meant. A computer program cannot be unambiguous. What may happen is that you interpret the program differently from the computer.
A further problem is the level of explicitness required. Machines do not have creative thinking, so every last little detail has to be spelled out. Humans, on the other hand, can guess, find connections, and derive on existing knowledge when interpreting a document. Excessive detail will only hamper understanding, which is directly contrary to computer programs. A computer will make all idiot mistakes you never thought anyone reviewing the program would.