Fermat said he had a proof.
Then when I reached college, I realized that many people had thought about the problem during the 18th and 19th centuries and so I studied those methods.
That particular odyssey is now over. My mind is now at rest.
Just because we can't find a solution it doesn't mean that there isn't one.
Always try the problem that matters most to you.
Mathematicians aren't satisfied because they know there are no solutions up to four million or four billion, they really want to know that there are no solutions up to infinity.