[seqfan] Re: Sequence related to jittery function

[bacher]

Sorry,

I should have thought things out a bit longer:
x=-1/2 is trivial since 2n+1=0 for $n=x=-1/2
and the second transformation is the almost equal
to the first (there remains some work to do but
I should pan out).


bacher <Roland.Bacher at univ-grenoble-alpes.fr> a écrit :

> Hi Jens,
> Try
>
> (a,b)=(38731904,40833784) for n=67108863
>
> for k=27
>
> (found randomly as follows:
>
> Consider a,b and x=n as variables and set up a cycle
> involving 27 of your two transformation different from 27
> iterations of the first formula and apply it to (a,b)
> getting (l_1,l_2) with l_1,l_2 two linear forms in a,b,x + constants.
> Solve the system for a=l_1,b=l_2 getting affine expressions
> with rational coefficients in terms of x.
> It seems that x=-1/2 is always a solution to both
> systems (I tried a few times with random compositions, this seems  
> also not to depend on the cycle length 27)
> Computing -1/2 modulo the numerator of the rational expression for a  
> or b yields n,
> plugging this values in the expressions for a and b, yields a,b.
>
> Check that it works using your formulae (it did with my solution,
> I hope I did not misstype it).
>
> I did not really think this over: all these things should be rather trivial
> to prove or this has to be a wonderful theory (I suspect that the
> first assertion is true)
>
> Best wishes Roland Bacher
>
> Jens Voß <jens at voss-ahrensburg.de> a écrit :
>
>> Let n be an integer greater than or equal to 2, and define
>>
>>   X_n := { (a, b) | 1 <= a < b <= n }
>>
>> to be the set of ordered pairs of distinct positive integers <= n.
>>
>> For every such pair (a, b) € X_n, let
>>
>>                  /  (b-a, b+a)      if  b+a <= n
>>   j_n ((a,b)) := |
>>                  \  (b-a, 2n+1-b-a) otherwise
>>
>> It is easy to see that
>>
>> (I)  j_n ((a,b)) € X_n for all (a, b) € X_n, i.e. j_n: X_n -> X_n and
>> (II) j_n is injective.
>>
>> Since X_n is finite, j_n is bijective, i.e. a permutation of X_n.
>>
>> I am interested in the cycle representation of j_n; in particular,
>> for every positive integer k, I would like to know the set of values
>> of n for which j_n contains a cycle of length k.
>>
>> It is not too hard to show that
>>
>> (1) j_n has a cycle of length 1 (i.e. a fixed point) iff n == 2 (mod 5).
>> (2) No n exists for which j_n has a cycle of length 2.
>> (3) j_n has a cycle of length 3 iff n == 3 (mod 7) or n == 6 (mod 13).
>>
>> I suppose that larger cycle lengths can be classified in a similar way,
>> but calculations quickly become rather tedious, so I used a simple
>> computer program to calculate the cycle lengths of j_n for all values
>> of n up to 3000 (after that, the program gets really slow).
>>
>> It appears that most smaller cycle lengths (except of course for 2) do
>> come up eventually, the first stubborn exception being the number 27.
>> However, the fact that this number (as well as the subsequent exceptions
>> 38, 41, 53, 57, 62, 98, 103, 122, ...) does not show up as a cycle
>> length of an j_n with n <= 3000 does not necessarily mean it can't do so
>> for some larger n.
>>
>> So the questions I am facing right now are: Is 27 a cycle length of an
>> f_n for some n? If so, what is the smallest such n? How about the other
>> exceptions I mentioned? How many numbers k > 2 do not occur as cycle
>> lengths; are there infinitely or finitely many or none at all?
>>
>> Does anyone have an idea on how to tackle these kind of problems? Has the
>> function j_n perhaps already been studied by someone - it does not look
>> too artificial to me!
>> Ideally, if every integer > 2 is in fact a cycle length of some j_n, I'd
>> like to submit the sequence of the smallest such n - currently, I am
>> limited to the values for 3 through 26 and don't even know whether it
>> can be extended beyond that point or whether it has a "hole" at k=27.
>>
>> Thanks and best regards
>> Jens
>>
>>
>>
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>
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