Generally discovering any kind of new math basically boils down to asking some question and hunting for an answer. A fun example might be something like the [Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems). This is a set of unsolved problems in math that each have a million dollar prize to anyone who can solve them. The Navier-Stokes problem, for example, is actually a really straightforward equation to write down if you know some basic physics, but it’s a differential equation, which isn’t necessarily very useful for many practical applications. And so the million dollar question is basically whether or not we can take this impractical equation and rewrite it as a practical one instead. [Numberphile did a video](https://www.youtube.com/watch?v=ERBVFcutl3M) on this topic if you’re interested.
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