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I tried reading it but could not understand the article (amazing, given its Quanta magazine). But I’m fascinated by the Fibonacci series so was compelled to ad
by Jun8 4y ago
I tried reading it but could not understand the article (amazing, given its Quanta magazine).
But I’m fascinated by the Fibonacci series so was compelled to add some comment. Here’s a cool fact that I recently learned F_gcd(m,n) = gcd(F_m, F_n). You can use this fact to prove that every prime divides a Fibonacci number (https://math.stackexchange.com/questions/695979/does-every-prime-divide-some-fibonacci-number/695984#695984 https://math.stackexchange.com/questions/695979/does-every-p...)
- svat 4y agoWhile this fact about gcds is true, your link goes to a proof (incidentally, happens to be mine) that does not use this fact. :-)
- Jun8 4y agoGreat to see you here and you're absolutely right: I got carried away with a comment (which, I now see, is yours, too :-) about Fibonacci numbers being a "strong divisibility sequence", this was the first time I've heard of this concept.
- svat 4y ago:-) Here's something else, almost entirely unrelated, that I like: we can prove that any two Fermat numbers 2^(2^n)+1 have gcd 1; this gives a proof (different from Euclid's) that there are infinitely many prime numbers: https://en.wikipedia.org/w/index.php?title=Fermat_number&oldid=1116879652#Basic_properties https://en.wikipedia.org/w/index.php?title=Fermat_number&old...
- bobbylarrybobby 4y agoI believe this also implies that the Fibonomial (not a typo) coefficients are integers, which is neat