The recurrence R(n) = R(n - 1) + gcd(n, R(n - 1)) generates primes. But why? It turns out it's essentially implementing trial division in disguise.
Previous video on this sequence: https://youtu.be/OpaKpzMFOpg
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Reference:
Eric Rowland, A natural prime-generating recurrence, Journal of Integer Sequences 11 (2008) 08.2.8 (13 pages).
https://cs.uwaterloo.ca/journals/JIS/VOL11/Rowland/rowland21.html
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0:00 Generating primes
1:26 Shortcut
4:42 What if 2 n - 1 is prime?
9:31 What if 2 n - 1 isn't prime?
14:46 Trial division
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Animated with Manim. https://www.manim.community
Music by Callistio.
Audio recorded at the Lawrence Herbert School of Communication at Hofstra University. https://www.hofstra.edu/communication/
Web site: https://ericrowland.github.io
Twitter: https://twitter.com/ericrowland