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Old 2019-08-15, 11:22   #23
AndrewWalker
 
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I've mentioned to Garambois via email that I've tried looking for small cycles of length 3 or more without success.
However, I had been using pari's factor() function in only the normal mode,so normal numbers would get a normal value of the sigma function.


Newer versions of pari allow this:


? factor(8,I)
%16 =
[ I 1]

[1 + I 6]



Giving a complex factorisation and a different value for sigma


I'm planning to do some searching with this over smaller ranges,


Andrew
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Old 2019-08-15, 12:39   #24
garambois
 
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Thank you Andrew for your help via email.

I finally managed to write successfully a program for the Sage software that runs and calculates sigma(z) for z a complex integer Gauss number.

I am looking for cycles of length c with 1<=c<=50 for all complex numbers z=a+b*I for 0<=a<=3000 and 0<=b<=3000 (a and b integers of course).
But it may take several years !
The program is extremely slow, because the terms of the complex aliquot sequence sometimes become difficult to factor after 40 iterations !
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Old 2019-08-16, 07:59   #25
AndrewWalker
 
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I'm changing a few pari scripts I wrote a while back and will give them a run. These will
mainly be randomised searches

One suggestion for anyone looking at these. The complex pairs I've found so far have
all had (1+I) , (1+2*I) or (1+4*I) as a factor.

So a search over a defined range (or a random search) could also work by multiplying
the value by one of these. Possibly to a small power. For two cycles this will greatly
improve the chance of success, for longer cycles???

garambois I'd also suggest running on a separate processor/computer a search just
up to 4 or even 6 length as it will get to larger values much quicker. If you do a 50 cycle
search, think carefully of what upper limit you will let these reach as some values might
increase rapidly just as in real numbers!

Andrew

Last fiddled with by AndrewWalker on 2019-08-16 at 08:00
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Old 2019-08-16, 09:11   #26
garambois
 
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Quote:
Originally Posted by AndrewWalker View Post

garambois I'd also suggest running on a separate processor/computer a search just
up to 4 or even 6 length as it will get to larger values much quicker. If you do a 50 cycle
search, think carefully of what upper limit you will let these reach as some values might
increase rapidly just as in real numbers!
Yes, I thought about it and it would be ideal !
Unfortunately my computer only has 12 threads.
And I have many more ideas to test.
I also want to calculate aliquot sequences starting on integer powers.

Maybe in a while....

I dream of a 120 thread computer !

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Old 2019-08-20, 11:15   #27
garambois
 
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Does anyone know if there exist Gauss untouchable numbers, with the sigma(z)-z function, for z a complex number, by analogy to the Erdös untouchable numbers with the sigma(n)-n function, with n an integer (https://oeis.org/A005114) ?

Of course, in the context of this question, for the calculation of sigma(z)-z, we consider the factorization of z with gaussian prime factors only taken in the first quadrant.
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Old 2019-10-22, 12:51   #28
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Paul ZIMMERMANN discovered a Gaussian Aliquot Cycle of length 6 !

-507253 + 70523 * I

Complete Gauss 6_cycle :

0 70523*I - 507253
1 335537*I - 225727
2 373843*I + 112387
3 130729*I + 243117
4 433407*I - 155165
5 294287*I + 139123

Congrats to Paul !

Can anyone confirm it, in addition to Paul and myself ?
To my knowledge, this is the first known Gaussian Aliquot Cycle of a length other than 2, isn't it !

I had told Paul about my research program for such cycles and I told him how slow it was.
He then decided to write a program in C much faster.
Here is the result after a few hours of operation of this new program !



Last fiddled with by garambois on 2019-10-22 at 13:46
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Old 2019-10-23, 06:51   #29
AndrewWalker
 
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Huge congratulations to Paul, amazed he did this so quick! I hope his code is much
quicker than what I've been doing in pari as it could then be quite useful!


Have confirmed it in pari, CSigma function is earlier in this thread:


(17:36) gp > a= 70523*I - 507253
%2 = -507253 + 70523*I
(17:37) gp > b=CSigma(a)-a
%3 = -225727 + 335537*I
(17:37) gp > c=CSigma(b)-b
%4 = 112387 + 373843*I
(17:37) gp > d=CSigma(c)-c
%5 = 243117 + 130729*I
(17:39) gp > e=CSigma(d)-d
%6 = -155165 + 433407*I
(17:39) gp > f=CSigma(e)-e
%7 = 139123 + 294287*I
(17:40) gp > CSigma(f)-f-a
%8 = 0


(17:40) gp > factor(a)
%9 =
[ 1 + I 1]

[ 21 + 26*I 1]

[ 18 + 35*I 1]

[268 + 63*I 1]

(17:46) gp > factor(b)
%10 =
[ 1 + I 1]

[ 62 + 3*I 1]

[ 4 + 5*I 1]

[653 + 302*I 1]

(17:46) gp > factor(c)
%11 =
[ 1 + I 1]

[243115 + 130728*I 1]

(17:46) gp > factor(d)
%12 =
[ -I 1]

[ 1 + I 1]

[ 1 + 2*I 1]

[ 29 + 4*I 1]

[2980 + 103*I 1]

(17:46) gp > factor(e)
%13 =
[ 1 + I 1]

[139121 + 294286*I 1]

(17:47) gp > factor(f)
%14 =
[ -I 1]

[ 1 + I 1]

[ 19 + 50*I 1]

[3272 + 2795*I 1]

(17:47) gp >

Again very nice hope this will result in many more to follow!
Maybe (1+I) * 1 or 2 gaussian primes is a good form to search?
(possibly with -I as the unit.) Are -1 and +I possible or not?



Today I finally reached 900 pairs in my 2 cycle search, will do
an updated list by or on the weekend.


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Old 2019-10-23, 08:31   #30
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Quote:
Originally Posted by AndrewWalker View Post
Today I finally reached 900 pairs in my 2 cycle search, will do
an updated list by or on the weekend.
Andrew
We are very interested in this list !
We have used your lists for our ongoing work...

Thank you for the confirmation of the discovery of the cycle of length 6 !
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Old 2019-10-28, 07:12   #31
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Here's the latest update, now 910 pairs! One pair from the Ranthony Ashley Clark thesis,
a (5,5) type pair had eluded me but I managed to find it earlier in the year by tweaking
one of my searches and going deeper on it than some others. Will follow with
a version showing factors.


Andrew
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File Type: txt GaussListMaster7.txt (38.2 KB, 32 views)
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Old 2019-10-28, 08:22   #32
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List with factors included, hopefully haven't goofed up, let me know if I have!


Andrew
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File Type: txt GaussListMaster7Fac.txt (152.3 KB, 34 views)
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Old 2019-11-02, 18:42   #33
garambois
 
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I just got back from vacation.
What a pleasant surprise !
Thank you very much for these new lists.
They will replace the old ones in our programs...



Paul Zimmermann's program is running right now.
No other cycle of length other than 2 until now, which would be added to the one of length 6 announced above.
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