[seqfan] Re: Puzzle of David Wilson

[Bob Selcoe]

Hi Matthijs and Seqfans,

Thanks for writing the paper and citing me, Matthijs!   I will need time to 
review.

One thing first - I'd be surprised if all real positive rational numbers < 1 
didn't create Wilson-chains.

Would you mind trying my "modular approach" with, say, 9/11?  It may require 
a somewhat different program - one that (I think) would eliminate many of 
the iterations required with the current program.

Here, the first term (11/2) would be the last term in your current program, 
then just make a "modular chain" until you reach 2/11.

So the first few steps are:

R_(1-9/11) = 11/2 = 121/22.   9/11 = 18/22.   121 mod 18 = 15.

 R_(15/22) = 22/15 = 242/165.   9/11 = 135/165.   242 mod 135 = 107.

R_(107/165) = 165/107 = 1815/1177.  9/11 = 963/1177.   1815 mod 963 = 852.
etc.

So generally, when a/b > 1/2, the process is first find b^2 mod a(b-a), = a' 
and b(b-a) = b', then continue the process with b*b' mod a*a' = a'' and b*b' 
= b'', etc., until reaching (b-a)/b.

(Of course, all fractions - and thus the a-primes and b-primes - may be in 
reduced form).

Certainly, this doesn't preclude other possible Wilson-chains existing for 
9/11 (as you've shown occurs with 7/9); but I think this particular approach 
can lead to more economical paths.  While it might take many steps, each 
step really is only one iteration; so I would think it wouldn't be too 
intense for the computer.  I would do this myself but I don't know the first 
thing about computer programming!

Best,
Bob S.



--------------------------------------------------
From: "Matthijs Coster" <seqfan at matcos.nl>
Sent: Wednesday, January 28, 2015 7:59 AM
To: "_Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Puzzle of David Wilson

> LS,
>
> For everybody who is interested in the puzzle of David Wilson (z -> z + 
> kr, or z -> 1/z).
> I wrote a paper with my results.
> See: http://www.matcos.nl/sequences/SeqFanPuzzle.pdf.
>
> Please send me your commands!
>
> Best regards,
>
> --Matthijs
>
>
>
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>