[r-t] Conjectures

[Philip Saddleton]

Richard Smith <richard at ex-parrot.com> wrote at 15:25:54 on Sun, 21 Nov 
2004
>If we believe that there are only a finite number of
>sporadics, then, for a given class of methods at a given
>parent stage there must be a stage beyond which no sporadic
>extensions exist.  (There are a finite number of methods
>each with a finite number of sporadic extensions, therefore
>there must be a "last" one.)  Let's call this the sporadic
>limit.  I hope it might be possible to obtain an upper bound
>on this sporadic limit.

My guess is that it is about 2n.

>You seem to be conjecting that, in a arithmetic extension
>series, each extension step above or below simply permutes
>some "coursing order".  Only (n-2) bells can be involved in
>this permutation.  To me, this suggests the maximum
>periodicity for Royal should be
>
>  lcm( 8.2.5, 6.3 ) = 2^4.3^2.5 = 720
>
>-- i.e. I think the factor of 3 from the step size and the
>factor of 3 from the permutation should both be preserved.

Yes - I should have included the step size as a possible separate 
factor.

>In the table, above, of periodicities identified, every
>divisor of 48 is present, except 48 itself and 1.  (Clearly
>1 would never be present.)  The absence of 48 is
>disappointing, but should perhaps not be taken as evidence
>that this theory is incorrect.  When I searched for
>extensions, I only looked as far as 60 bells.  For many
>high-period extensions, the parent method has itself been
>sporadic (i.e. not part of the arithmetic series) -- in the
>case of period 24 extensions, every example had a sporadic
>parent.  It seems quite plausible that if such an extension
>exists, the arithmetic region might not have begun by 60
>bells.

A further conjecture: there are no sporadic extensions beyond the start 
of the arithmetic region. Do you have an example of an extension that 
includes both sporadic (other than the parent) and arithmetic elements?

>I would go further and say "no factors that are prime powers
>exceeding the n-2 where n is the number of working bells".
>This is borne out by the absence of any factors of 5 in any
>of the minor periodicities I found.

This seems reasonable for TD methods, since only n-2 bells can be 
permuted by the repeated sections, but could there be examples of a TP 
or Alliance method with 4 blows behind at the half-lead?

-- 
Regards
Philip
http://www.saddleton.freeuk.com/