[r-t] Conjectures

[Richard Smith]

Philip Saddleton wrote:

> a) Stages at which an extension of a method exists are a finite set,
> plus an infinite arithmetic series (possibly omitting stages where the
> cycle lengths have certain factors - these in turn form infinite
> series).

I was just thinking along these lines myself!

Like you, I'm very sceptical that an extension might exist
sporadically (i.e. not as part of an infinite arithmetic
series) at an infinite number of stages.

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.

Similarly, there must be a maximum periodicity for the
arithmetic extensions for a given class of methods at a
given stage.  We have touched upon this before.  On six
bells, I identified examples with the following
periodicities:

  2  = 2
  4  = 2^2
  6  = 2.3
  8  = 2^3
  12 = 2^2.3
  16 = 2^4
  24 = 2^3.3

And, on 15th Aug, you said:

| I would imagine it might be possible to contrive a Royal
| method extending by 4 places below the treble (swapping 2
| and rotating 5) and by 6 places above (rotating 3), where
| the gap is 16x3x5 = 240.

We know that the maximum step size (i.e. length of repeating
section -- so a BC extension is 2, and a BCDE is 4) for the
extension of the above or below work is (n-2).  If we have 8
and 6 above and below, straight away this gives a
periodicity of lcm(6,8) = 2^3.3 = 24, and this is without
considering the need for regular lead heads.

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.

What about Minor?  How does this theory compare with the
periodicities I actually found?

The permutations on the "coursing order" can only be of
orders 1, 2, 3 or 4, and the step size can only be 2 or 4.
This suggests a maximum periodicity of

  lcm(4.4, 4.3) = 2^4.3 = 48.

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.

Alternatively, it might be that the particular 778
parent methods that I studied did not contain an example,
but a larger selection of methods might have turned one up.

You also conject:

> The step in the series has no factors that are prime
> powers exceeding the number of working bells.

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.

Richard