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I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.
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.
I know it's a rare privilege, but if one can really tackle something in adult life that means that much to you, then it's more rewarding than anything I can imagine.
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.
Mathematics... is a bit like discovering oil. ... But mathematics has one great advantage over oil, in that no one has yet ... found a way that you can keep using the same oil forever.
I was so obsessed by this problem that I was thinking about it all the time - when I woke up in the morning, when I went to sleep at night - and that went on for eight years.