period

[Don Reble]

> I conjectured that the following two sequences are cyclic.
>
>   x(n)=[a*x(n-1)+b]/p^r,
>   a, b are real number, [x] is integer part of x,
>   p is prime, p^r is the highest power of p dividing [a*x(n-1)+b]
>
>   seq.1.    x(0), p, a, b = 1, 2, 2.00013, 3.0
>   seq.2.    x(0), p, a, b = 1, 2, 2.00001, 3.2
>
> ...please calculate the periods and tell me them.

(I edited out some apparent transmission errors.)


The first sequence has a period of 1790641. I had wondered whether it
would ever repeat: the sequence grows unsteadily until the 477470'th
term, when it reaches the value
    82721263883011610886526961867779718135764506625125
    11743699901796233367506991160435951413702199247578
    48336819257023283527033772942121308102778129423389
    10053098213870566201892389659879944244624498503906
    76244208391103283037041703648964775995893652810637
    75443503638514561304340346005699508846909678163541
    30869180866795754559855095373101163156336097219855
    71841626091479071631154016283655936808751927734552
    38107621661269300592877592978724640932952936515280
    23530580303113261797777433800708978953387458811266
    82954756815787375433632658164067205860448170308260
    27,
a somewhat non-descript 552-digit number.

The second sequence is easy: it never goes past 
    81104929169215939436320658138839143546777420460293
    79282485589496928568880735463557636136597534817627
    54702800422427897905708391641024017523200893908310
    4625
and has a fundamental period of 180492.

--
Don Reble       djr at nk.ca