Hearted Youtube comments on Another Roof (@AnotherRoof) channel.
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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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You may have already seen it, but Mark Rober released a YT Short a few weeks ago based on this video! It's called "The REAL Truth Behind the DVD Logo".
It seems likely that they used this vid in particular, because they seem to pull the number 58460 out of nowhere, which you calculate at the end by "meticulously" counting frame-by-frame to get x=316, y=185. They also give the same answer of 16m 14s.
Although, they apparently didn't realise that these numbers only apply to one specific simulation uploaded on YouTube and wouldn't work for a normal DVD player! Even though you mention this immediately before giving your calculations. So that's a bit embarrassing.
And of course, they unfortunately don't give credit to this channel, and their video now has over 50,000,000 views.
I really love your videos, and this one specifically has helped me recently with a project I'm working on!
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