No one in the mathematical world believes that Fermat actually had a valid proof of his famous Last Theorem (click here for the Wikipedia article).
But I’d be interested to see what his non-proof looked like.
Looking at why something is broken is often a good way to get insight into a new research direction. Think of how the mistake in Euler’s proof of Fermat’s theorem for n=3 led to the discovery of the concept of unique factorization: click here to read about this in the interesting Fermat’s Last Theorem blog).
Does anyone know of attempts to best-guess Fermat’s non proof?
Here’s one effort
After a bit of googling I came across the following website which claims to give an elementary proof: click here to see it.
Fermat's Last Spreadsheet 