[math-fun] Generalized Fibonacci sequences and arithmetic progressions
Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it? Jim Propp
Is this statement even true of the classic Fibonacci sequence? Can you show me a remainder class that the Fibonacci sequence avoids forever? Mod 2: 0, 1 ... We have already hit both classes. Mod 3: 0, 1, 1, 2 ... Already hit all three. Mod 4: 0, 1, 1, 2, 3 ... done. Mod 5: 0, 1, 1, 2, 3, 0, 3, 3, 1, 4 ... OK, that took a little while. Mod 6: 0, 1, 1, 2, 3, 5, 2, 1, 3, 4 ... I seem to recall doing this for many many moduli when I was in high school, and finding empirically that the Fibonacci sequence eventually visits all remainder classes. I would be surprised to see a counterexample. On Tue, Aug 8, 2017 at 10:15 PM, James Propp <jamespropp@gmail.com> wrote:
Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Never mind. I don't know what I was remembering. The Fibonacci sequence certainly avoids 8n+4 and 8n+6. On Tue, Aug 8, 2017 at 10:36 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Is this statement even true of the classic Fibonacci sequence? Can you show me a remainder class that the Fibonacci sequence avoids forever?
Mod 2: 0, 1 ... We have already hit both classes. Mod 3: 0, 1, 1, 2 ... Already hit all three. Mod 4: 0, 1, 1, 2, 3 ... done. Mod 5: 0, 1, 1, 2, 3, 0, 3, 3, 1, 4 ... OK, that took a little while. Mod 6: 0, 1, 1, 2, 3, 5, 2, 1, 3, 4 ...
I seem to recall doing this for many many moduli when I was in high school, and finding empirically that the Fibonacci sequence eventually visits all remainder classes. I would be surprised to see a counterexample.
On Tue, Aug 8, 2017 at 10:15 PM, James Propp <jamespropp@gmail.com> wrote:
Given a two-sided generalized Fibonacci sequence ...,n-m,m,n,m+n,... (with m,n in Z and not both zero), must there exist a two-sided arithmetic progression ...,a-d,a,a+d,... (with a,d in Z and d nonzero) that is disjoint from it?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
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Allan Wechsler -
James Propp