Youtube comments of Thomas Morgan (@mrtthepianoman).
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Here's my (arguably) simpler proof of the first problem. It's a proof by induction. Base case: the pair 1, 2 works (easy to find). Induction step: assume we have found a pair a, b such that a < b and a | b^2 +1 and b | a^2 +1. Then we can write b^2 + 1 = a * c for some positive integer c. Then it is clear that c | b^2 + 1. Thus it suffices to show that b | c^2 + 1. But c = (b^2 + 1) / a, so c^2 +1 = (b^4 + 2b^2 + 1) / a^2 + 1 = (b^4 + 2b^2 + a^2 + 1) / a^2. We know b | a^2 + 1, so b divides the numerator. But if b | a^2 + 1, then b is relatively prime to a^2 (the denominator). It follows that b divides the quotient, and we are done.
Thus, starting from any pair (a, b) we can find, we can generate an infinite sequence as follows. Start with (a, b). b^2 + 1 = a * c gives us the pair (b, c). c^2 + 1 = b * d gives us the pair (c, d). And so on. This proof is nice because it does not require any knowledge about properties of Fibonacci numbers. And it follows easily from the pattern of the first few examples that you found (and they were the first few examples I found as well).
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I think the breakdown in this video captures it pretty well. There are some areas of mathematics that have practical value. In such areas, any solution will do. However, there are some areas in mathematics that are only interesting to pure mathematicians. In such areas, the mathematicians desire beauty, understanding, and insight. I think chess can be like this. To some, chess is about winning or losing. If that is the way you think about it, then chess AI plays the game really well. However, for some the way the game is played makes a difference in their enjoyment of the game. Knowing a bunch of chess theory and seeing it play out is part of what makes a game exciting and fun, so a seemingly random sequence of moves would make for a worse game, even if it results in a win
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