20210331, 15:36  #1046  
"Garambois JeanLuc"
Oct 2011
France
585_{10} Posts 
Quote:
Thank you very much Oliver ! I manage to get the program to work and it seems to work even with bases that are not prime numbers. The program is remarkably fast. I did several tests. I will be doing more extensive testing in the next few days. I don't know yet how to do it myself, but if you have a little time, it would be very useful for our work, please, to add a few lines of code so that at the end, the program shows us on a single line the complete divisor in the form of a product, like this : "d = p_{1}^a_{1} * p_{2}^a_{2} * p_{3}^a_{3} ..." which is the product of all the prime factors found taking into account their multiplicity. So, it will be very easy to test the abundance of this divisor d. Many thanks for all ! 

20210331, 16:04  #1047  
"Oliver"
Sep 2017
Porta Westfalica, DE
497_{10} Posts 
You are welcome.
Quote:
Quote:
But for small exponents, we could have a full factorization. But in this case, yafu will be much more efficient in factoring such numbers. Last fiddled with by kruoli on 20210331 at 16:31 Reason: Additional information. Removed bogus words. Spelling. 

20210331, 16:15  #1048  
"Garambois JeanLuc"
Oct 2011
France
1111_{8} Posts 
Quote:
Of course, I understood that your program is to be used only for very very large exponents. And of course, we will never have full factorization in these extreme cases. For much smaller exponents, there will be yafu ... 

20210331, 17:19  #1049 
"Garambois JeanLuc"
Oct 2011
France
3^{2}·5·13 Posts 
Sorry, I just realized that I forgot to attach the file in post #1045 !
Here it is ... 
20210331, 20:03  #1050 
"Oliver"
Sep 2017
Porta Westfalica, DE
7×71 Posts 
This is a new version of my program with the output format suggested by garambois. To return to the old format, add v as the first argument.

20210401, 17:44  #1051  
"Garambois JeanLuc"
Oct 2011
France
3^{2}×5×13 Posts 
Quote:
OK, many thanks Oliver. Everything seems to work perfectly ! I will be doing some testing over the next week and will let you know if I have any problems. Just a detail, I am unable to run the program with the v argument. But I don't necessarily think I need it, because the new display of the result suits me much more. Your program makes it possible to have all the prime numbers even up to 10 ^ 9 in a few tens of minutes if the base is not too large and that with exponents of the order of 1,000,000 ! And I tried a very large exponent for base 3 : 3 ^ 5 * 5 ^ 3 * 7 ^ 2 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47 and we are very very far from having an abundant d !!! Another detail : it seems impossible to enter the exponent in the form of a product of prime factors as in the example above, it must be calculated before to have the exponent in the form of a single number to be able to enter it in the program. Very often, when we want to try exponents, we enter them as products of prime numbers, because we "construct" them. But this is really a detail and it is very fast to build the exponent with another program in parallel ;) 

20210401, 21:39  #1052  
"Oliver"
Sep 2017
Porta Westfalica, DE
7·71 Posts 
Quote:
Quote:
Quote:
Last fiddled with by kruoli on 20210401 at 21:40 Reason: Grammar. 

20210402, 13:27  #1053 
"Garambois JeanLuc"
Oct 2011
France
3^{2}×5×13 Posts 
Everything seems to be working very, very well.
On the other hand, this leads me to ask you a question before continuing my tests : Please, what is the size limit for the exponents ? Because for bases 3 and 5 it seems extremely, but then really extremely difficult to find an exponent "i" that assures us an abundant s(3^i) or s(5^i). After an hour of different tests, I entered these parameters in the program : Code:
mono patf.exe v 3 "3 ^ 20 * 5 ^ 18 * 7 ^ 16 * 11 ^ 14 * 13 ^ 12 * 17 ^ 10 * 19 ^ 8 * 23 ^ 6 * 29 ^ 4 * 31 ^ 2 * 37 * 41 * 43 * 47 * 53 * 59 * 61 * 67 * 71 * 73 * 79 * 83 * 89 * 97" 100000 It was then that I realized that this exponent had 129 digits ! It's hard to believe that the program can handle exponents of this size ! Note : I must limit the last parameter of the program to 100,000, otherwise it becomes time consuming to test the abundance of d with Sage. But it would seem that it is not very useful to consider larger prime numbers : it does not influence too much the abundance of the number ... 
20210402, 14:27  #1054 
"Oliver"
Sep 2017
Porta Westfalica, DE
7·71 Posts 
The exponent is not limited in size by my program. Instead, it is limited by Microsoft's (or the mono team's) implementation of BigInteger. Modular exponentation is extremely efficient, especially for small modulos. And our modulos are quite small. I ran some test to show you why the size does not matter as much as you think:
Code:
ModPow(2, 10^2, 1277) took 0 ms. ModPow(2, 10^4, 1277) took 0 ms. ModPow(2, 10^8, 1277) took 0 ms. ModPow(2, 10^16, 1277) took 0 ms. ModPow(2, 10^32, 1277) took 0 ms. ModPow(2, 10^64, 1277) took 0 ms. ModPow(2, 10^128, 1277) took 0 ms. ModPow(2, 10^256, 1277) took 0 ms. ModPow(2, 10^512, 1277) took 0 ms. ModPow(2, 10^1024, 1277) took 0 ms. ModPow(2, 10^2048, 1277) took 0 ms. ModPow(2, 10^4096, 1277) took 0 ms. ModPow(2, 10^8192, 1277) took 0 ms. ModPow(2, 10^16384, 1277) took 1 ms. ModPow(2, 10^32768, 1277) took 2 ms. ModPow(2, 10^65536, 1277) took 5 ms. ModPow(2, 10^131072, 1277) took 10 ms. ModPow(2, 10^262144, 1277) took 20 ms. ModPow(2, 10^524288, 1277) took 39 ms. ModPow(2, 10^1048576, 1277) took 82 ms. ModPow(2, 10^2, 2^311) took 0 ms. ModPow(2, 10^4, 2^311) took 0 ms. ModPow(2, 10^8, 2^311) took 0 ms. ModPow(2, 10^16, 2^311) took 0 ms. ModPow(2, 10^32, 2^311) took 0 ms. ModPow(2, 10^64, 2^311) took 0 ms. ModPow(2, 10^128, 2^311) took 0 ms. ModPow(2, 10^256, 2^311) took 0 ms. ModPow(2, 10^512, 2^311) took 0 ms. ModPow(2, 10^1024, 2^311) took 0 ms. ModPow(2, 10^2048, 2^311) took 0 ms. ModPow(2, 10^4096, 2^311) took 0 ms. ModPow(2, 10^8192, 2^311) took 0 ms. ModPow(2, 10^16384, 2^311) took 1 ms. ModPow(2, 10^32768, 2^311) took 2 ms. ModPow(2, 10^65536, 2^311) took 6 ms. ModPow(2, 10^131072, 2^311) took 12 ms. ModPow(2, 10^262144, 2^311) took 23 ms. ModPow(2, 10^524288, 2^311) took 47 ms. ModPow(2, 10^1048576, 2^311) took 95 ms. ModPow(2, 10^2, 2^631) took 0 ms. ModPow(2, 10^4, 2^631) took 0 ms. ModPow(2, 10^8, 2^631) took 0 ms. ModPow(2, 10^16, 2^631) took 0 ms. ModPow(2, 10^32, 2^631) took 0 ms. ModPow(2, 10^64, 2^631) took 0 ms. ModPow(2, 10^128, 2^631) took 0 ms. ModPow(2, 10^256, 2^631) took 0 ms. ModPow(2, 10^512, 2^631) took 0 ms. ModPow(2, 10^1024, 2^631) took 0 ms. ModPow(2, 10^2048, 2^631) took 0 ms. ModPow(2, 10^4096, 2^631) took 0 ms. ModPow(2, 10^8192, 2^631) took 1 ms. ModPow(2, 10^16384, 2^631) took 2 ms. ModPow(2, 10^32768, 2^631) took 4 ms. ModPow(2, 10^65536, 2^631) took 10 ms. ModPow(2, 10^131072, 2^631) took 24 ms. ModPow(2, 10^262144, 2^631) took 43 ms. ModPow(2, 10^524288, 2^631) took 97 ms. ModPow(2, 10^1048576, 2^631) took 169 ms. The computation of 10^{n} takes way longer than the modular exponentiation in my test (not shown here). 
20210402, 19:50  #1055  
"Alexander"
Nov 2008
The Alamo City
261_{16} Posts 
Quote:


20210403, 14:12  #1056  
"Garambois JeanLuc"
Oct 2011
France
3^{2}×5×13 Posts 
Quote:
OK, thank you very much Alexander for all of your help. The page is finally accessible from my website and I am very happy to have this new page which recapitulates all the conjectures ! To access, click here : http://www.aliquotes.com/conjectures_mersenneforum.html Please, would it be possible for an administrator to add a link to this page in the very first post of the topic, a sentence like that, or something better formulated for an English speaker ;) : "Access the regularly updated page which summarizes all the conjectures published on this topic by clicking here". 

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