How do I find the period of repetition of this modular equation?

Question

EDITED QUESTION:

I have this equation $$\left(x+\frac{2}{3}\right)\,mod\, \frac{13}{5}$$

and I want to find period after which its value repeats.

It can be written as $$\left(\frac{3x+2}{3}\right)\,-\, \frac{13}{5}q = 0$$

I am equating it to some number (here I am equating it to zero to make the calculations simple), in order to find its period. And $q$ is the quotient of the mod process.

It can be simplified as $$15x+10=39q$$ It can be re-written as $$(15x+10) \,mod \, 39$$

Its period is 39, but when I checked programmatically, its period is 13. I want the smallest interval as the period. Where is the mistake? Can someone help me?

Answer 1

First of all, your computations are exact : the smallest period is $36$.

It is of interest to make a zoom on a smaller part. We can observe:

• that the curve is made of parabolic arcs, looking very irregular at this scale. Note that the range of values is $[0,1/3].$

• that, there is a rather unexpected phenomenon ; the local (aka as "windowed") frequency is increasing till value 4.5 ; there is a big contrast between "just before" 4.5 and "just after", this value, a little as if the "aperiodic behavior" is all concentrated before this value and a kind of "reset" is done once value $4.5$ has been crossed. A further analysis is necessary.

Matlab program that has generated the figure:

x=35.:0.0001:37.5;
fr=@(x,y)(x-y.*floor(x./y));
u=fr(fr(9*x/4,(3/5)).*fr(2*x/3,3),1/3);
plot(x,u);