3 years, 86 days

## size of exported plot is not deformed...

When I open the exported svg-file the height-width ratio is not correct.

I have no Idea how to solve this ...

## exportplot...

I have seen this answer very late ...

It's very use full for my App :)

thank you :)

## starup region...

Good morning acer,

So I can put all the big code in the startup region and have just the Buttons and plot-window in front.

I will work with that.

Yes you are right with randomize(). It's also in my main-code. I just leave it out in the example.

ok this helps.

I understand

## App...

Thanks you

I have written a short example-file for testing this.

pb-App_0.1.mw

The first Click on the Button gives an error message in maplePlayer. with the second click it works. The error is why ?

Now my code for producing the coordinates (A) in above file is very large and I can't insert it all the "edit click code" of the button.

What will be  best way to do that ?

## {seq}(seq(g^i mod 78, i= 1..12), g= G); ...

The bases not generated by your {seq}... are bases of order 2 and 6 eccept the square bases 25,49.

n := 78;

co := coprimes(n);
co := {1, 5, 7, 11, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 67, 71, 73, 77}
g1 := {seq}(seq(g^i mod n, i = 1 .. per), g = G);
g1 := {1, 5, 7, 11, 19, 25, 31, 37, 41, 43, 47, 49, 55, 59, 61, 67, 71, 73}
co minus g1;
{17, 23, 29, 35, 53, 77}

g1 includes the squares 25 and 49.

Numbers coprime to their Euler totients are sometimes called  cyclic numbers  (A003277)  because a group whose order is such a number must be cyclic.  The number 2 is cyclic but all the other cyclic numbers are odd.  Thus, the main conjecture is that  2  is the only cyclic number  without Carmichael multiples.  This can be stated even more compactly: Any odd cyclic number has Carmichael multiples.

Does this has to to with the bases of order 2 ?

and here

it stands out that  g(78) is 6 with g = number of groups of order n (A000001).

g(n) = 2   if  n  is  either  the square of a prime  or  a squarefree number with  only one  of its prime factors congruent to  1  modulo another  (A054395).

If g1 had not the squares 25 and 49 the Set difference of g1 and  co namely

{17, 23, 29, 35, 53, 77}   belongs to all bases of order 2 and 6.

## mod rule...

this is fantastic :)

So if this is a clear criterion (the pair mod rule is like a bisection) then it should be possible to find all pairs and so all bases for one period (or order) without other criterions. Intelligent going through all the coprimes of a number (e.g. 78). Starting with one known base, storing the partner base in a table with cache or remember and  so at the end having all the bases.

Do you think this is an alternive way for finding the bases for a given period/order?

## bases...

good morning :)

I have read your last message in the attached file I have calculated  the example.

CL_base_finding.mw

The missing base of the last row {seq}... are of period 2 and 6.

I have not found a way to solve this until yet, maybe later ...

Maybe this helps:

for our example n=78:

These are base-pairs which occur clearly in may plots inside  a period. Each pair has mirrored remainder sequences. The first half remainder sequence for the 1st base is the second half of the 2nd base. I dont know If this is usefull for you somehow.

After looking 3 weeks on that I have found an easy way to find the pairs, but not found an mod rule to calculated it.

The first 3 element are for base 2. Here are no pairs.

{{25}, {53}, {77}, {5, 47}, {7, 67}, {11, 71}, {17, 23}, {19, 37}, {29, 35}, {31, 73}, {41, 59}, {43, 49}, {55, 61}}

Please print this Array, then you can see the mirrored remainders:

Matrix(23, 3, [[2, 25, [1, 25]], [2, 53, [1, 53]], [2, 77, [1, 77]], [3, 55, [1, 55, 61]], [3, 61, [1, 61, 55]], [4, 5, [1, 5, 25, 47]], [4, 31, [1, 31, 25, 73]], [4, 47, [1, 47, 25, 5]], [4, 73, [1, 73, 25, 31]], [6, 17, [1, 17, 55, 77, 61, 23]], [6, 23, [1, 23, 61, 77, 55, 17]], [6, 29, [1, 29, 61, 53, 55, 35]], [6, 35, [1, 35, 55, 53, 61, 29]], [6, 43, [1, 43, 55, 25, 61, 49]], [6, 49, [1, 49, 61, 25, 55, 43]], [12, 7, [1, 7, 49, 31, 61, 37, 25, 19, 55, 73, 43, 67]], [12, 11, [1, 11, 43, 5, 55, 59, 25, 41, 61, 47, 49, 71]], [12, 19, [1, 19, 49, 73, 61, 67, 25, 7, 55, 31, 43, 37]], [12, 37, [1, 37, 43, 31, 55, 7, 25, 67, 61, 73, 49, 19]], [12, 41, [1, 41, 43, 47, 55, 71, 25, 11, 61, 5, 49, 59]], [12, 59, [1, 59, 49, 5, 61, 11, 25, 71, 55, 47, 43, 41]], [12, 67, [1, 67, 43, 73, 55, 19, 25, 37, 61, 31, 49, 7]], [12, 71, [1, 71, 49, 47, 61, 41, 25, 59, 55, 5, 43, 11]]])

## Totient(Totient)...

my attention is intuitively going to that row of your code.

Now I have read your algotithm-explaination  obove and will see if I find a way to solve it.

## this is very intuitive: it could be tha...

this is very intuitive:

it could be that it has to to with the first totient of this code-part

TT:= Totient(Totient(n))

good morning,

I have tried to find a rule for the behavior of your code.

The only clear rule is: it seems to work for all odd periods.

For many or most of even periods the list of bases is not complete.

The funny thing is, odd periods produce much more intersting plots in my program.

So I will use this part (bases for odd periods) of your code.

best regards

## orders are correct...

you are right, in base 2 and base Carmichael(n)/2 there are missing bases.

For my work this is not important because I only use periods between 6 and 24.

So I am happy to use the code furthermore :)

## Algorithm...

this is good to know :)

I have worked with your fast base-Algorithm and the speedincreasing is very helpfull for me.

I have not checked your bases-result with Multiplicative-Order but I will do now.

I will now read your Error-Description and come back the next days.

Best regards :)

## Worksheet...

I am working with Maple 2021.

But I have just tried with the mw.

Its running now :)

I come back when I have studied this code.

Thanks again

## complete code...

Good morning,

this code will help me to calculate faster :)

Actually I have copied it to Maple-Worksheet and after running I get this error-message:

Error, invalid `\$` operator

right below the module-code.

May you send it again as a worksheet-file (mw).

I will work through the stuff to understand it.

Good weekend

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