What The Fuck is a Pell’s Equation?
This might be the most useless thing you can learn about, but it’s undeniably interesting. Pell’s equation is an equation of the form x^2 - ny^2 = 1 (I'm pissed that I can’t use Latex). In the example above, we discuss the case where n = 2.
How Do You Solve One?
Sure, you can read all about this equation here, but this post discusses how fucking cool the proof is. I didn’t come up with this btw, because I’m a moron.
In How to Solve It by George Pólya, one of my favorite books about Mathematics and problem solving in general, one of the claims is that:
If you can’t solve a problem, there is a simpler problem that you can’t solve. Find it.
The Pell’s equation is hard as fuck to solve. The proof above started out by finding a simple solution.The trivial solution x = 1 or x = -1 and y = 0 is easy to notice, but not interesting (just like me). The solution x= 3 and y = 2 would be the next pair of numbers that works if you are just plugging in positive integers. From the analysis attached, we notice a pattern in the pair of solution. So we grab on to it like Rose to that fucking door in the Titanic and never let go. We boldly make a claim (bde) which applies to ALL solution of the Pell’s Equation and prove it using induction.
This is an example of effective problem-solving. We start out with a guess, analyze the results of that guess, notice a pattern, and prove that the pattern applies to every solution to the problem.