x->x*ceiling(x)

[Roland Bacher]

> On Thu, 29 Aug 2002, David Boyd wrote:
> 
> > In answer to John's question, the result about 14,23 is in
> > my paper in Acta Arith, v. 24 (1977), 89--98.   Shortly
> > thereafter, I was able to show that starting with any of
> > the following gives a Pisot sequence which is not linearly
> > recurrent:  7,15   8,14   9,15  9,17  9,19  10,18  10,21.
> > David Cantor suggested that probably 4,13  also is a bad
> > starting pair.   One of my old Macs did a computation
> > verifying this but since it ran for a few months I wouldn't
> > claim that this has yet been proved, but if true it is
> > a matter of a finite computation.
> 
>    Could you briefly explain how you prove such things
> (and in particular, why they reduce to finite computations)?
> 
> > I don't think the sequences generated by  x -> x*ceil(x)
> > have much to do with Pisot sequences....
> 
>    I didn't think there was much real resemblance either -
> just a reminiscence, so to speak.  
> 
>    Regards, John Conway
> 

I cannot prove that the sequence gotten by iterating 
x->x*ceil(x) and starting with x_0=1+1/n   gets always integral,
but it does so for n=1,....,100.

the first integral indices are:

0, 1, 2, 3, 18, 2, 3, 4, 6, 7, 26, 4, 9, 3, 4, 8, 6, 4, 56, 11, 3, 4, 42, 4, 33, 7, 5, 4, 38, 5, 79, 6, 4, 15, 14, 8, 200, 29, 13, 5, 36, 3, 4, 5, 7, 10, 11, 8, 6, 20, 47, 27, 43, 9, 41, 9, 10, 23, 37, 17, 18, 6, 7, 6, 32, 15, 225, 7, 73, 11, 20, 12, 182, 9, 16, 7, 10, 15, 196, 8, 11, 62, 23, 5, 26, 4, 5, 8, 85, 11, 18, 61, 22, 177, 59, 10, 187, 10, 20, 6

local maxima are achieved for n=1,2,3,4,5 (18), 11 (26), 19 (56), 37 (200) and
67 (225)  .

hence x_{18} \in N for x_0=1+1/5;

I have computed these values by doing all computations over the integers
(multiply by n) and by truncating modulo n^250. This avoids the explosion
of the integers (of order 2^(2^k) after $k$ iterations) 
and gives the correct answer if the final index i(n) is < 250 
(or perhaps 249 or 248). If the algorithm does not stop befor 245 you should
increase precision (work with n^500 or even higher).

Roland Bacher