Then look up Wiles' proof of the Taniyama-Shimura conjecture.
The math required is so advanced, and there have been numerous people who taught they solved it but turned out to be wrong.
It's extremely unlikely that Fermat had solved it.
PS. He wrote that note when he was young (and working on special cases, small n and such) and probably realized later on that he didn't have a proof for the general case.
Edit: There was no letter to a friend. It was in the margin of a book he was reading. More like a little reminder to himself to remember to jot down the proof.
Your financial situation is part of a much bigger problem you are trying to solve.
How to have a good life. Think you should optimize with that in mind.
Completely agree with the diminishing return comment. If social interactions/relationships are lacking in your life, you'll get a very high return for your time spent.
(Obviously extremely easy to say to you typing on my iPhone. What has always worked for me is taking the smallest possible step and turning that into a habit.)
With regards to clothing you can use instagram to follow men or pages whose style you like and use that as inspiration.
Finally, I'd recommend getting in above average shape. Make going to the gym a part of your life. An hour everyday before work of straight after.
While you're doing some/all of these, I'd start to just talk to everyone. Men, women, old and young. That way you get into the habit of becoming more sociable and confident.
I know these tips may sound superficial, but they help shifting an external locus of control into an internal one.