Fermat's Last Theorem.

[doodab]

But if that is the case why hasn't anyone at least come up with a plausible "flawed" proof to show what Fermat might have done. His proof must have appeared to be good at least superficially.

There are thousands of flawed proofs. Some partial ones as well.

[BlasterBates]

There we go

Simple proof of Fermat's last theorem

the wonders of the Internet

[NotAllThere]

But if that is the case why hasn't anyone at least come up with a plausible "flawed" proof to show what Fermat might have done. His proof must have appeared to be good at least superficially.

There have been various explanations. He did prove it for n = 4. He probably had in mind generalising that for higher powers.

[doodab]

For simple proofs that aren't actually wrong you can start here:

Proof of Fermat's Last Theorem for specific exponents - Wikipedia, the free encyclopedia

I got to the point of only needing to prove it for prime exponents when I was at uni. I personally think there is some mileage in looking at logarithms.