4 ms·
Since the Fibonacci sequence preserves limits, the adjoint functor theorem says there must be some function G such that n divides F(m) if and only if G(n) divid
by OscarCunningham 5y ago
Since the Fibonacci sequence preserves limits, the adjoint functor theorem says there must be some function G such that n divides F(m) if and only if G(n) divides m. This is called the sequence of entry points for the Fibonacci sequence, because G(n) is given by the first m such that n divides F(m) [https://oeis.org/A001177 https://oeis.org/A001177].
Of course since G is a left adjoint it has the dual property that lcm(G(n),G(m)) = G(lcm(n,m)).
Can we do more advanced category theory here? What is the Kan extension of the Fibonacci sequence along the Mersenne numbers?
- ogogmad 5y agoI was thinking about finding a left adjoint G. I see you've proven its existence using a sledgehammer: the adjoint functor theorem. Your explicit construction for G(n) as the smallest m such that n divides F(m) is very informative. It goes well with the idea that an adjoint is somehow an optimiser. Thanks. That entry point sequence is weird.