Is there something special about the Fibonacci numbers, or does it work for, say, any function that grows exponentially?
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Replying to @metaweta @InertialObservr
If it works for 2^n, I'll eat my hat.
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Replying to @johncarlosbaez @InertialObservr
Oh! Hahaha, yeah, that wouldn't work at all.
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Replying to @metaweta @InertialObservr
What I'm reminded of is this: the probability of two large integers being relatively prime is 6/pi^2: https://en.wikipedia.org/wiki/Coprime_integers#Probability_of_coprimality … You have to define what you mean here, of course.
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Replying to @johncarlosbaez @InertialObservr
And that makes me think it has nothing to do with Fibonacci numbers, per se, but rather that the series grows in the right way and doesn't have any special interaction with the primes. Powers of 2 only use the prime 2, but the Fibonacci series probably hits them all.
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Replying to @metaweta @InertialObservr
Sounds right, but it could take work to show Fibonacci numbers "don't have any special interaction with the primes". Some "probably true" things of this sort are very hard to prove. And our host has not pointed us to a proof.
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Ask and ye shall receivehttps://math.stackexchange.com/questions/3390098/proof-that-the-ratio-between-the-logs-of-the-product-and-lcm-of-the-fibonacci-nu …
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