[r-t] Conjectures

[Richard Smith]

Philip Saddleton wrote:
> Richard Smith wrote:

> >I hope it might be possible to obtain an upper bound
> >on this sporadic limit.
>
> My guess is that it is about 2n.

I have appended a list of all the extensions I found.  As
you can see, I found examples of a sporadic extension of
minor on 22 bells.  One such method is Neath Delight
(&-34-1-2-23-4-5,1) extended with 2DE/1DEFG to produce:

  &-34-1-2-23-4-5-6-67-8-9-0-0E-T-A-B-BC-D-F-G-GH-J-K,1

(I have checked this example on up to 254 bells to confirm
that it is not, say, an example of a period 48 arithmetic
series.)

I think I'm more pessimistic about the position of the
sporadic limit.  I could believe that a carefully
constructed example might extend sporadically about as far
as the maximum periodicity for the stage -- 48 for
extensions of Minor.

Take the extension of London Minor that gives London Major
as an example (2DE/2EF).  When N=0 (mod 6) (where N is the
number of bells), we find that at sufficiently large stages,
the pivot bell required to make the extension regular is the
(N/3-6)th.  Similarly the actual pivot bell is bound above
by (2N-42)/9.  These lines actually intersect at N=12, which
would initially seem to confirm your "2n" hypothesis.  In
practice, we need to be a little more careful (not in the
least because these formulae are yielding negative numbers
at this point!).

The fact 2, 3, 5 and 7 appear in the wrong place in the
coursing order (because they don't start by hunting four
blows to a fish tail), means that you have to deal with
these bells separately.  The odd bells turn up in the above
formulae as negative numbers, so I would say that until both
N/3-6 > 2 and (2N-42)/9 > 2 there is potential for sporadic
extensions (even though none in fact exist).  This works out
as N > 30, which is consistent with the sporadic extension
of Neath (to 22) and with the conjecture that things should
have settled down by aroud the maximum periodicity.

> A further conjecture: there are no sporadic extensions beyond the start
> of the arithmetic region.

I could imagine an extension of Minor that, say,
sporadically worked on 8 bells and then on 6n bells.  Would
you say that the arithmetic region started at 6 or at 12?
If we say that the arithmetic region only begins at the
first extension (rather than at the parent), I'm inclined to
agree.

Of course, if we did discover, say, an extension of Minor
that worked sporadically on 14 bells and arithmetically on
2n bells (including on 12), I would probably redefine the
arithmetic region to start at 18 and say that 12 was
sporadic, so the conjecture would remain true ;-)

> Do you have an example of an extension that
> includes both sporadic (other than the parent) and arithmetic elements?

No.  See the table below.

> >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?

Fair point.  But even this only gives "no ... prime powers
exceeding n-1".

Richard

------------------------------------------------------------

Table of extension sequences up to 60 bells for regular
TDMMs ordered by frequency (including differential hunters
and methods with 1 lead courses)

   Freq  Stages (excluding parent method)
   1519  8
   1059  10, 14, 18, ... 58
    645  10
    540  8, 10, 12, ... 60
    100  12, 18, 24, ... 60
     58  18, 30, 42, 54
     50  8, 12, 16, ... 60
     38  14, 22, 30, ... 54
     27  16, 28, 40, 52
     21  12
     18  8, 20, 32, 44, 56
     17  14
      9  10, 16, 22, ... 58
      8  10, 34, 58
      7  8, 10
      6  26, 50
      5  28, 52
      5  18
      4  22
      4  12, 20, 28, ... 60
      4  10, 22, 34, 46, 58
      3  8, 12
      2  20, 44
      2  14, 26, 38, 50
      2  10, 18, 26, ... 58
      1  8, 14, 20, ... 56
      1  22, 38, 54
      1  10, 12, 14, ... 60
   ----
   4156