[QUOTE=kruoli;574996]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. [/QUOTE]


Thank you very much Oliver for those explanations.
I fear that my essays on bases 3, 5, 17 and especially 21 (what a crazy base !) will cause me to reach the limits of my computer hardware and my software.
I think I will give you a little report in the next few days about these tests ...

[QUOTE=garambois;575069]. . .
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 [URL="http://www.aliquotes.com/conjectures_mersenneforum.html"]clicking here[/URL]".[/QUOTE]See if what I added will work for you.

[QUOTE=EdH;575082]See if what I added will work for you.[/QUOTE]


PERFECT !
Many thanks Edwin !

My attempt to get the first termination in column 9 of base 10 instead produced the fourth merge in that column. According to Ed's script (and verified manually in FactorDB):

[code]10^109:i3390 merges with 1691544:i83[/code]

[QUOTE=kruoli;574965]If somebody is eager to implement it with primesieve and GMP, this will run in seconds for sure! I updated my code with multithreading, so it should be somewhat faster (it will scale about linear to the available physical cores).[/QUOTE]

I am currently in the process of porting it. I hope to have it done in the next couple of days.

[QUOTE=kruoli;574965]If somebody is eager to implement it with primesieve and GMP, this will run in seconds for sure! I updated my code with multithreading, so it should be somewhat faster (it will scale about linear to the available physical cores).

Yes, I did not plan for that originally. It simply expected a string of decimal digits. Now, with this version, you can enter e.g. [C]patf 3 "3 ^ 5 * 5 ^ 3 * 7 ^ 2 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47" 1000000000[/C]. The quotation marks are important here! And only the exponent can be in this form. Please have no value start with the digit 0. BigInteger of C# handles such numbers octally by default.[/QUOTE]

[QUOTE=Happy5214;575109]I am currently in the process of porting it. I hope to have it done in the next couple of days.[/QUOTE]

The port is done. If that command is the invocation you tested, it definitely didn't just take seconds. My Kubuntu Focus ran it in 1 minute and 58 seconds. Instead of attaching it here, I'll link to the copy in my GitHub repository ([url]https://github.com/happy5214/rps-scripts/blob/main/aliquotPowers/powerTrialFactoring.cpp[/url]) so I can update it as necessary.

It seems like I was too optimistic for the amount of time savable. On my Laptop, my code will run the command in around two minutes using eight threads (i7-3630QM), the sieving alone takes half a minute. Primesieve only takes about a second on my machine up to 1,000,000,000.

Maybe your program can also be multithreaded? Ignore that if you already have planned that. And maybe one could also examine the abundance inside the program automatically? I saw that your sigma-function is not used in the code in the moment.

[QUOTE=Happy5214;575099]My attempt to get the first termination in column 9 of base 10 instead produced the fourth merge in that column. According to Ed's script (and verified manually in FactorDB):

[code]10^109:i3390 merges with 1691544:i83[/code][/QUOTE]


OK, I will add this merge in the next update. Thank you very much.

[QUOTE=kruoli;575170]And maybe one could also examine the abundance inside the program automatically ?[/QUOTE]

Yes, that would be very interesting, if it is not too complicated ?

Because I'm having trouble testing the abundance of very large divisors d !
You will be able to see in my next post which will be published in a few minutes or hours ... I am waiting for a program to finish running.

[QUOTE=kruoli;575170]It seems like I was too optimistic for the amount of time savable. On my Laptop, my code will run the command in around two minutes using eight threads (i7-3630QM), the sieving alone takes half a minute. Primesieve only takes about a second on my machine up to 1,000,000,000.

Maybe your program can also be multithreaded? Ignore that if you already have planned that. And maybe one could also examine the abundance inside the program automatically? I saw that your sigma-function is not used in the code in the moment.[/QUOTE]

I've posted a multi-threaded version (using pthreads, so use Linux or WSL) to that link. The same computer (Kubuntu Focus) runs the same test using 8 threads in 20.75 seconds. I'm running short on time today, so I'll get to the abundance later. You could copy the code from some of my older programs (which can be found in the same GitHub repo).

I would like to share with you here some results that I obtained after several hours of testing with Oliver's program.
My first work was to try out several odd bases (prime numbers and composite numbers) with a (huge) exponent constructed in a very stupid way :
[CODE]i = 3 ^ 100 * 5 ^ 99 * 7 ^ 98 * 11 ^ 97 * 13 ^ 96 * 17 ^ 95 * 19 ^ 94 * 23 ^ 93 * 29 ^ 92 * 31 ^ 91 * 37 ^ 90 * 41 ^ 89 * 43 ^ 88 * 47 ^ 87 * 53 ^ 86 * 59 ^ 85 * 61 ^ 84 * 67 ^ 83 * 71 ^ 82 * 73 ^ 81 * 79 ^ 80 * 83 ^ 79 * 89 ^ 78 * 97 ^ 77 * 101 ^ 76 * 103 ^ 75 * 107 ^ 74 * 109 ^ 73 * 113 ^ 72 * 127 ^ 71 * 131 ^ 70 * 137 ^ 69 * 139 ^ 68 * 149 ^ 67 * 151 ^ 66 * 157 ^ 65 * 163 ^ 64 * 167 ^ 63 * 173 ^ 62 * 179 ^ 61 * 181 ^ 60 * 191 ^ 59 * 193 ^ 58 * 197 ^ 57 * 199 ^ 56 * 211 ^ 55 * 223 ^ 54 * 227 ^ 53 * 229 ^ 52 * 233 ^ 51 * 239 ^ 50 * 241 ^ 49 * 251 ^ 48 * 257 ^ 47 * 263 ^ 46 * 269 ^ 45 * 271 ^ 44 * 277 ^ 43 * 281 ^ 42 * 283 ^ 41 * 293 ^ 40 * 307 ^ 39 * 311 ^ 38 * 313 ^ 37 * 317 ^ 36 * 331 ^ 35 * 337 ^ 34 * 347 ^ 33 * 349 ^ 32 * 353 ^ 31 * 359 ^ 30 * 367 ^ 29 * 373 ^ 28 * 379 ^ 27 * 383 ^ 26 * 389 ^ 25 * 397 ^ 24 * 401 ^ 23 * 409 ^ 22 * ​​419 ^ 21 * 421 ^ 20 * 431 ^ 19 * 433 ^ 18 * 439 ^ 17 * 443 ^ 16 * 449 ^ 15 * 457 ^ 14 * 461 ^ 13 * 463 ^ 12 * 467 ^ 11 * 479 ^ 10 * 487 ^ 9 * 491 ^ 8 * 499 ^ 7 * 503 ^ 6 * 509 ^ 5 * 521 ^ 4 * 523 ^ 3 * 541 ^ 2 * 547 * 557 * 563 * 569 * 571 * 577 * 587 * 593 * 599 * 601 * 607 * 613 * 617 * 619 * 631 * 641 * 643 * 647 * 653 * 659 * 661 * 673 * 677 * 683 * 691 * 701 * 709 * 719 * 727 * 733 * 739 * 743 * 751 * 757 * 761 * 769 * 773 * 787 * 797 * 809 * 811 * 821 * 823 * 827 * 829 * 839 * 853 * 857 * 859 * 863 * 877 * 881 * 883 * 887 * 907 * 911 * 919 * 929 * 937 * 941 * 947 * 953 * 967 * 971 * 977 * 983 * 991 * 997 [/CODE]The entry into Oliver's program has been the same each time :
[B]mono patf.exe -v b "i" 100000[/B]
With of course the number representing the base in place of the character "b" and the product of prime numbers above in place of the character "i".
Below are the results obtained (Only prime factors < 1000 of d have been displayed here for clarity, but it is very easy to get them with Oliver's program in a matter of minutes) :
[CODE][B]base 3 : abundance : 0.719[/B] with d = 11^99 * 13^97 * 23^94 * 47^88 * 59^86 * 71^83 * 83^80 * 107^75 * 109^74 * 131^71 * 167^64 * 179^62 * 181^61 * 191^60 * 227^54 * 229^53 * 239^51 * 251^49 * 263^47 * 277^44 * 311^39 * 313^38 * 347^34 * 359^31 * 383^27 * 419^22 * 421^21 * 431^20 * 433^19 * 443^17 * 467^12 * 479^11 * 491^9 * 503^7 * 541^3 * 563^2 * 587^2 * 599^2 * 601^2 * 647^2 * 659^2 * 683^2 * 709^2 * 719^2 * 733^2 * 743^2 * 757^2 * 827^2 * 829^2 * 839^2 * 863^2 * 887^2 * 911^2 * 947^2 * 971^2 * 983^2 ...
[B]base 5 : abundance : 0.667[/B] with d = 11^98 * 19^95 * 31^92 * 59^86 * 71^83 * 79^81 * 101^77 * 109^74 * 131^71 * 139^69 * 149^68 * 151^67 * 179^62 * 181^61 * 191^60 * 199^57 * 211^56 * 239^51 * 251^49 * 269^46 * 271^45 * 311^39 * 331^36 * 359^31 * 379^28 * 389^26 * 401^24 * 409^23 * 419^22 * 431^20 * 439^18 * 461^14 * 479^11 * 491^9 * 499^8 * 541^3 * 569^2 * 571^2 * 599^2 * 619^2 * 631^2 * 659^2 * 691^2 * 719^2 * 739^2 * 751^2 * 811^2 * 829^2 * 839^2 * 859^2 * 911^2 * 919^2 * 941^2 * 971^2 * 991^2 ...
[B]base 7 : abundance : 1.358[/B] with d = 3^100 * 19^95 * 29^93 * 31^92 * 37^91 * 47^88 * 59^86 * 83^80 * 103^76 * 109^74 * 131^71 * 139^69 * 167^64 * 199^57 * 223^55 * 227^54 * 251^49 * 271^45 * 283^42 * 307^40 * 311^39 * 367^30 * 383^27 * 389^26 * 419^22 * 439^18 * 467^12 * 479^11 * 503^7 * 523^4 * 563^2 * 587^2 * 607^2 * 613^2 * 619^2 * 643^2 * 647^2 * 653^2 * 691^2 * 701^2 * 709^2 * 719^2 * 727^2 * 757^2 * 787^2 * 809^2 * 811^2 * 839^2 * 859^2 * 877^2 * 887^2 * 971^2 * 983^2 ...
[B]base 11 : abundance : 1.104[/B] with d = 5^99 * 7^99 * 19^95 * 43^89 * 79^81 * 83^80 * 107^75 * 127^72 * 131^71 * 139^69 * 151^67 * 157^66 * 167^64 * 211^56 * 227^54 * 239^51 * 263^47 * 271^45 * 283^42 * 307^40 * 317^37 * 347^34 * 359^31 * 389^26 * 397^25 * 421^21 * 431^20 * 439^18 * 479^11 * 491^9 * 503^7 * 523^4 * 547^2 * 563^2 * 571^2 * 607^2 * 659^2 * 661^2 * 739^2 * 743^2 * 757^2 * 787^2 * 797^2 * 811^2 * 827^2 * 829^2 * 887^2 * 919^2 * 967^2 ...
[B]base 13 : abundance : 1.220[/B] with d = 3^100 * 23^94 * 43^89 * 53^87 * 61^85 * 79^81 * 103^76 * 107^75 * 127^72 * 131^71 * 139^69 * 179^62 * 181^61 * 191^60 * 199^57 * 211^56 * 251^49 * 263^47 * 283^42 * 311^39 * 337^35 * 347^34 * 367^30 * 389^26 * 419^22 * 439^18 * 443^17 * 467^12 * 491^9 * 503^7 * 523^4 * 547^2 * 563^2 * 571^2 * 599^2 * 607^2 * 647^2 * 659^2 * 677^2 * 701^2 * 719^2 * 727^2 * 751^2 * 757^2 * 797^2 * 823^2 * 859^2 * 883^2 * 887^2 * 907^2 * 911^2 * 919^2 * 971^2 * 991^2 * 997^2 ...
base 15 : abundance : 0.707 with d = 3 * 11^98 * 59^86 * 71^83 * 109^74 * 131^71 * 179^62 * 181^61 * 191^60 * 239^51 * 251^49 * 311^39 * 353 * 359^31 * 419^22 * 431^20 * 479^11 * 491^9 * 541^3 * 599^2 * 659^2 * 719^2 * 829^2 * 839^2 * 911^2 * 971^2 ...
[B]base 17 : abundance : 0.508[/B] with d = 19^95 * 43^89 * 47^88 * 59^86 * 67^84 * 83^80 * 103^76 * 127^72 * 149^68 * 151^67 * 157^66 * 179^62 * 191^60 * 223^55 * 229^53 * 239^51 * 251^49 * 263^47 * 271^45 * 293^41 * 307^40 * 331^36 * 359^31 * 383^27 * 389^26 * 409^23 * 433^19 * 443^17 * 463^13 * 467^12 * 491^9 * 509^6 * 523^4 * 563^2 * 587^2 * 599^2 * 613^2 * 631^2 * 647^2 * 659^2 * 727^2 * 739^2 * 757^2 * 773^2 * 829^2 * 859^2 * 863^2 * 883^2 * 919^2 * 967^2 * 971^2 ...
[B]base 19 : abundance : 1.184[/B] with d = 3^100 * 31^92 * 59^86 * 67^84 * 71^83 * 79^81 * 101^77 * 103^76 * 107^75 * 127^72 * 149^68 * 151^67 * 157^66 * 167^64 * 179^62 * 211^56 * 223^55 * 227^54 * 229^53 * 233^52 * 277^44 * 307^40 * 331^36 * 349^33 * 379^28 * 383^27 * 397^25 * 431^20 * 439^18 * 461^14 * 487^10 * 523^4 * 547^2 * 557^2 * 563^2 * 599^2 * 607^2 * 613^2 * 653^2 * 659^2 * 683^2 * 701^2 * 709^2 * 743^2 * 751^2 * 787^2 * 811^2 * 821^2 * 827^2 * 839^2 * 863^2 * 887^2 * 907^2 * 911^2 * 971^2 * 983^2 * 991^2 * 997^2 ...
base 21 : abundance : 0.170 with d = 47^88 * 59^86 * 83^80 * 109^74 * 131^71 * 167^64 * 227^54 * 251^49 * 311^39 * 383^27 * 419^22 * 467^12 * 479^11 * 503^7 * 563^2 * 587^2 * 647^2 * 709^2 * 719^2 * 757^2 * 839^2 * 887^2 * 971^2 * 983^2 ...
[B]base 23 : abundance : 0.934[/B] with d = 7^99 * 11^97 * 19^95 * 29^93 * 43^89 * 67^84 * 79^81 * 83^80 * 103^76 * 107^75 * 173^63 * 191^60 * 197^58 * 199^57 * 227^54 * 233^52 * 251^49 * 263^47 * 269^46 * 277^44 * 283^42 * 317^37 * 349^33 * 359^31 * 367^30 * 379^28 * 383^27 * 419^22 * 431^20 * 461^14 * 467^12 * 479^11 * 503^7 * 509^6 * 523^4 * 541^3 * 563^2 * 571^2 * 619^2 * 631^2 * 643^2 * 659^2 * 727^2 * 743^2 * 751^2 * 787^2 * 827^2 * 829^2 * 839^2 * 877^2 * 907^2 * 911^2 * 919^2 * 971^2 * 983^2 * 997^2 ...
[B]base 29 : abundance : 0.839[/B] with d = 7^98 * 13^97 * 23^94 * 59^86 * 67^84 * 71^83 * 83^80 * 103^76 * 107^75 * 139^69 * 149^68 * 151^67 * 167^64 * 173^63 * 179^62 * 181^61 * 197^58 * 199^57 * 223^55 * 227^54 * 239^51 * 283^42 * 347^34 * 373^29 * 383^27 * 397^25 * 419^22 * 431^20 * 439^18 * 463^13 * 487^10 * 499^8 * 521^5 * 523^4 * 547^2 * 571^2 * 587^2 * 631^2 * 643^2 * 647^2 * 683^2 * 691^2 * 701^2 * 709^2 * 719^2 * 787^2 * 811^2 * 821^2 * 863^2 * 883^2 * 919^2 * 991^2 ...
[B]base 31 : abundance : 1.952[/B] with d = 3^100 * 5^99 * 11^98 * 23^94 * 43^89 * 79^82 * 83^80 * 101^77 * 127^72 * 139^69 * 149^68 * 151^67 * 167^64 * 179^62 * 199^57 * 223^55 * 239^51 * 251^49 * 263^47 * 271^45 * 317^37 * 331^36 * 347^34 * 349^33 * 367^30 * 383^27 * 397^25 * 421^21 * 463^13 * 487^10 * 491^9 * 499^8 * 523^4 * 571^2 * 587^2 * 617^2 * 619^2 * 631^2 * 643^2 * 647^2 * 653^2 * 719^2 * 733^2 * 739^2 * 743^2 * 787^2 * 823^2 * 827^2 * 853^2 * 859^2 * 863^2 * 877^2 * 883^2 * 911^2 * 947^2 * 967^2 * 983^2 * 991^2 ...
base 33 : abundance : 0.512 with d = 3 * 83^80 * 107^75 * 131^71 * 167^64 * 227^54 * 239^51 * 263^47 * 347^34 * 359^31 * 421^21 * 431^20 * 479^11 * 491^9 * 503^7 * 563^2 * 659^2 * 743^2 * 757^2 * 827^2 * 829^2 * 887^2 ...
[B]base 37 : abundance : 1.767[/B] with d = 3^100 * 7^99 * 11^98 * 41^90 * 47^88 * 67^84 * 71^83 * 73^82 * 83^80 * 101^77 * 107^75 * 127^72 * 139^69 * 149^68 * 151^67 * 157^66 * 197^58 * 211^56 * 223^55 * 229^53 * 233^52 * 263^47 * 269^46 * 271^45 * 307^40 * 359^31 * 367^30 * 379^28 * 397^25 * 419^22 * 443^17 * 491^9 * 571^2 * 599^2 * 613^2 * 619^2 * 659^2 * 691^2 * 719^2 * 733^2 * 739^2 * 743^2 * 751^2 * 773^2 * 787^2 * 811^2 * 823^2 * 839^2 * 863^2 * 877^2 * 887^2 * 971^2 * 983^2 ...
base 45 : abundance : 0.703 with d = 3 * 11^98 * 59^86 * 71^83 * 109^74 * 131^71 * 179^62 * 181^61 * 191^60 * 239^51 * 251^49 * 311^39 * 359^31 * 419^22 * 431^20 * 479^11 * 491^9 * 541^3 * 599^2 * 659^2 * 719^2 * 829^2 * 839^2 * 911^2 * 971^2 ...
[B]base 79 : abundance : 1.707[/B] with d = 3^100 * 7^100 * 13^96 * 43^89 * 47^88 * 59^86 * 71^83 * 101^77 * 103^76 * 107^75 * 127^72 * 139^69 * 191^60 * 199^57 * 211^56 * 227^54 * 251^49 * 271^45 * 277^44 * 281^43 * 307^40 * 311^39 * 317^37 * 331^36 * 337^35 * 359^31 * 379^28 * 389^26 * 397^25 * 419^22 * 421^21 * 443^17 * 457^15 * 463^13 * 491^9 * 503^7 * 541^3 * 587^2 * 593^2 * 607^2 * 619^2 * 631^2 * 647^2 * 659^2 * 691^2 * 739^2 * 757^2 * 773^2 * 821^2 * 823^2 * 827^2 * 859^2 * 877^2 * 883^2 * 941^2 * 947^2 * 983^2 * 991^2 * 997^2 ...
[B]base 4007 : abundance : 0.862[/B] with d = 11^98 * 13^97 * 19^95 * 31^92 * 47^88 * 61^85 * 67^84 * 83^80 * 89^79 * 101^77 * 103^76 * 107^75 * 127^72 * 131^71 * 137^70 * 151^67 * 163^65 * 181^61 * 211^56 * 233^52 * 239^51 * 263^47 * 271^45 * 307^40 * 349^33 * 379^28 * 397^25 * 419^22 * 431^20 * 439^18 * 461^14 * 463^13 * 479^11 * 487^10 * 491^9 * 503^7 * 521^5 * 547^2 * 587^2 * 599^2 * 601^2 * 619^2 * 643^2 * 661^2 * 719^2 * 727^2 * 811^2 * 823^2 * 827^2 * 859^2 * 911^2 * 919^2 * 947^2 * 967^2 * 983^2 * 991^2 * 997^2 ...
base 4849845 : abundance : 1.052 with d = 3^4 * 5 * 7 * 1429 * 2939 * 3251 * 11677 * 13859 * 18191 * 27011 * 31391 * 35279 * 48179 * 49451 * 63839 * 66491 * 70619 * 85931 * 87491 * 88079 * 96851 * 97499 * 97919
[B]base 10000000019 : abundance : 1.169[/B] with d = 7^99 * 11^98 * 13^97 * 19^96 * 23^94 * 29^93 * 71^83 * 83^80 * 107^75 * 109^74 * 131^70 * 137^70 * 139^69 * 157^66 * 163^65 * 181^61 * 191^60 * 211^56 * 229^53 * 233^52 * 263^47 * 269^46 * 271^45 * 277^44 * 331^36 * 359^31 * 397^25 * 419^22 * 439^18 * 461^14 * 467^12 * 479^11 * 487^10 * 491^9 * 509^6 * 521^4 * 523^4 * 547^2 * 607^2 * 619^2 * 647^2 * 683^2 * 691^2 * 701^2 * 709^2 * 719^2 * 727^2 * 733^2 * 743^2 * 751^2 * 787^2 * 811^2 * 821^2 * 827^2 * 829^2 * 859^2 * 863^2 * 883^2 * 907^2 * 919^2 * 941^2 * 947^2 * 971^2 ...[/CODE]Note that prime numbers appear in bold.
With this exponent i, it is only for bases which are prime numbers that we have an abundance and for a base which is a large primorial without the factor 2 :
for 7 (we already knew it for 7 thanks to warachwe, post [URL]https://www.mersenneforum.org/showpost.php?p=574385&postcount=1022[/URL]), for primes 11, 13, 31, 37, 79, 10000000019 and
for 4849845 = 3 * 5 * 7 * 11 * 13 * 17 * 19.
This result is already remarkable in itself and shows the power of Oliver's program !
And now, we know the exponent i, such that for other bases b than 7 which are prime numbers, we have abundant s(b^i).
And now I also know that I don't need to let Happy's program run to test all the odd exponents for bases 3 and 5 to find the smallest exponent that will give an s(3^i) and an s(5^i) abundant.
I have no chance of succeeding !
On the other hand, I let the Happy's program run for base 7 : there, I have a chance to find the smallest exponent which will give an abundance.
For base 3, Happy's program had tested all odd exponents up to 6,000,000 and for base 5, up to 5,000,000. For base 7, I am at 4,600,000 and the program is therefore still running.
For bases which are not prime numbers, this exponent i chosen above is extremely bad in trying to prove the abundance of s(b^i).
Indeed, we know for example that s(15^35) is abundant, but our disproportionately large exponent i does not give an abundant s(15^i) !
It surprises me a lot !
Note also that the more the odd base seems to be the product of a large number of small prime factors (primorial base without the factor 2), the less d seems to have prime factors !
It also surprises me a lot !

It is therefore not a good idea to take a stupidly chosen exponent "i" like the one above.
I made a lot of other attempts to find another exponent i that would assure me an abundant s(3^i).
But as soon as i exceeds a certain size, it becomes impossible for me to test the abundance of d.
And if we consider an exponent i which is of the order of magnitude of 3000# (product of all the prime factors from 3 to 2999) by adding exponents to the small prime factors, Oliver's program seems to find its limits by announcing a "segmentation error".

Let's try to do better for base 3.
If I refer to the explanations of warachwe ([URL]https://www.mersenneforum.org/showpost.php?p=573720&postcount=983[/URL]), I can attempt to build a more efficient i exponent for base 3.
So I built the following exponent j which is much more efficient for the base 3 :
[CODE]j = 3^7 * 5^5 * 7^4 * 11^3 * 13^3 * 17^2 * 19^2 * 23^2 * 29^3 * 31^2 * 37^2 * 41^2 * 43^2 * 47^2 * 53^2 * 59 * 61 * 67 * 71^2 * 73^2 * 79 * 83^2 * 89 * 97^2 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 167 * 173 * 179 * 181 * 191 * 193 * 197 * 199 * 211 * 223 * 227 * 229 * 233 * 239 * 241 * 251 * 257 * 263 * 269 * 271 * 277 * 281 * 283 * 293 * 307 * 311 * 313 * 317 * 331 * 337 * 347 * 349 * 353 * 359 * 367 * 373 * 379 * 383 * 389 * 397 * 401 * 409 * 419 * 421 * 431 * 433 * 439 * 443 * 449 * 457 * 461 * 463 * 467 * 479 * 487 * 491 * 499 * 503 * 509 * 521 * 523 * 541 * 547 * 557 * 563 * 569 * 571 * 577 * 587 * 593 * 599 * 601 * 607 * 613 * 617 * 619 * 631 * 641 * 643 * 647 * 653 * 659 * 661 * 673 * 677 * 683 * 691 * 701 * 709 * 719 * 727 * 733 * 739 * 743 * 751 * 757 * 761 * 769 * 773 * 787 * 797 * 809 * 811 * 821 * 823 * 827 * 829 * 839 * 853 * 857 * 859 * 863 * 877 * 881 * 883 * 887 * 907 * 911 * 919 * 929 * 937 * 941 * 947 * 953 * 967 * 971 * 977 * 983 * 991 * 997 * 1009 * 1013 * 1019 * 1021 * 1031 * 1033 * 1039 * 1049 * 1051 * 1061 * 1063 * 1069 * 1087 * 1091 * 1093 * 1097 * 1103 * 1109 * 1117 * 1123 * 1129 * 1151 * 1153 * 1163 * 1171 * 1181 * 1187 * 1193 * 1201 * 1213 * 1217 * 1223 * 1229 * 1231 * 1237 * 1249 * 1259 * 1277 * 1279 * 1283 * 1289 * 1291 * 1297 * 1301 * 1307 * 1319 * 1321 * 1327 * 1361 * 1367 * 1373 * 1381 * 1399 * 1409 * 1423 * 1427 * 1429 * 1433 * 1439 * 1447 * 1451 * 1453 * 1459 * 1471 * 1481 * 1483 * 1487 * 1489 * 1493 * 1499 * 1511 * 1523 * 1531 * 1543 * 1549 * 1553 * 1559 * 1567 * 1571 * 1579 * 1583 * 1597 * 1601 * 1607 * 1609 * 1613 * 1619 * 1621 * 1627 * 1637 * 1663 * 1667 * 1669 * 1693 * 1697 * 1699 * 1709 * 1721 * 1723 * 1733 * 1741 * 1747 * 1753 * 1759 * 1777 * 1783 * 1789 * 1801 * 1811 * 1823 * 1831 * 1847 * 1861 * 1867 * 1871 * 1873 * 1877 * 1879 * 1889 * 1901 * 1907 * 1913 * 1931 * 1933 * 1951 * 1973 * 1979 * 1987 * 1993 * 1999 * 2003 * 2011 * 2017 * 2027 * 2039 * 2053 * 2063 * 2069 * 2083 * 2087 * 2089 * 2099 * 2113 * 2129 * 2141 * 2143 * 2153 * 2161 * 2179 * 2203 * 2207 * 2213 * 2237 * 2239 * 2251 * 2267 * 2269 * 2273 * 2281 * 2287 * 2297 * 2309 * 2311 * 2333 * 2339 * 2341 * 2351 * 2357 * 2371 * 2377 * 2381 * 2383 * 2389 * 2393 * 2399 * 2411 * 2423 * 2437 * 2441 * 2447 * 2459 * 2467 * 2473 * 2477 * 2503 * 2531 * 2539 * 2543 * 2549 * 2551 * 2557 * 2579 * 2591 * 2593 * 2609 * 2617 * 2621 * 2657 * 2659 * 2663 * 2671 * 2677 * 2683 * 2687 * 2689 * 2693 * 2699 * 2707 * 2711 * 2713 * 2719 * 2729 * 2731 * 2741 * 2753 * 2767 * 2777 * 2789 * 2791 * 2797 * 2801 * 2803 * 2819 * 2833 * 2837 * 2851 * 2857 * 2887 * 2897 * 2903 * 2909 * 2927 * 2939 * 2953 * 2957 * 2963 * 2969 * 2999 * 3001 * 3011 * 3019 * 3023 * 3041 * 3049 * 3061 * 3067 * 3083 * 3109 * 3137 * 3163 * 3167 * 3169 * 3187 * 3203 * 3209 * 3217 * 3221 * 3229 * 3251 * 3253 * 3257 * 3259 * 3271 * 3299 * 3301 * 3307 * 3313 * 3319 * 3323 * 3329 * 3331 * 3343 * 3359 * 3361 * 3389 * 3391 * 3407 * 3413 * 3433 * 3449 * 3457 * 3463 * 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31469 * 31649 * 31721 * 31793 * 31799 * 31859 * 32003 * 32009 * 32141 * 32159 * 32381 * 32531 * 32561 * 32573 * 32633 * 32771 * 32789 * 32843 * 32933 * 33023 * 33053 * 33119 * 33179 * 33191 * 33461 * 33479 * 33521 * 33569 * 33623 * 33713 * 33749 * 33773 * 33809 * 33941 * 34253 * 34283 * 34319 * 34439 * 34631 * 34883 * 34913 * 34949 * 35069 * 35081 * 35099 * 35111 * 35291 * 35573 * 35831 * 35933 * 35993 * 35999 * 36083 * 36191 * 36251 * 36353 * 36383 * 36479 * 36563 * 36629 * 36761 * 36791 * 36821 * 36923 * 36929 * 37013 * 37049 * 37139 * 37181 * 37253 * 37379 * 37619 * 37853 * 37871 * 37991 * 38039 * 38183 * 38189 * 38201 * 38231 * 38303 * 38333 * 38453 * 38459 * 38501 * 38639 * 38669 * 38723 * 38861 * 38873 * 38891 * 38933 * 39089 * 39233 * 39239 * 39419 * 39443 * 39521 * 39551 * 39569 * 39659 * 39779 * 39953 * 39971 * 39983 * 39989 * 40193 * 40283 * 40343 * 40559 * 40763 * 40823 * 40853 * 40949 * 41081 * 41231 * 41243 * 41381 * 41399 * 41603 * 41609 * 41621 * 41669 * 41729 * 41969 * 42023 * 42071 * 42089 * 42131 * 42221 * 42359 * 42473 * 42611 * 42719 * 42743 * 42821 * 42923 * 43013 * 43313 * 43391 * 43541 * 43649 * 43661 * 43691 * 43721 * 43793 * 43943 * 44111 * 44129 * 44189 * 44249 * 44273 * 44501 * 44543 * 44651 * 44699 * 44729 * 44879 * 44909 * 45053 * 45119 * 45131 * 45179 * 45263 * 45329 * 45569 * 45599 * 45641 * 45971 * 46181 * 46199 * 46229 * 46349 * 46523 * 46589 * 46619 * 46643 * 46691 * 46703 * 46751 * 47189 * 47279 * 47363 * 47501 * 47513 * 47543 * 47609 * 47639 * 47741 * 48029 * 48131 * 48221 * 48239 * 48413 * 48479 * 48563 * 48593 * 48731 * 48761 * 49103 * 49193 * 49253 * 49331 * 49433 * 49463 * 49481 * 49499 * 49559 * 49811 * 49853 * 49919 * 50021 * 50051 * 50261 * 50273 * 50411 * 50423 * 50513 * 50591 * 50741 * 50873 * 50969 * 50993 * 51203 * 51503 * 51521 * 51539 * 51659 * 51893 * 52103 * 52121 * 52163 * 52289 * 52361 * 52379 * 52511 * 52553 * 52571 * 52583 * 52631 * 52733 * 52883 * 53051 * 53093 * 53309 * 53411 * 53453 * 53549 * 53591 * 53639 * 53849 * 53951 * 54011 * 54101 * 54251 * 54293 * 54401 * 54413 * 54443 * 54773 * 54941 * 54959 * 55229 * 55439 * 55469 * 55631 * 55661 * 55673 * 55721 * 55733 * 55799 * 55829 * 55889 * 55931 * 56009 * 56081 * 56099 * 56123 * 56393 * 56489 * 56519 * 56531 * 56663 * 56681 * 56783 * 56891 * 56909 * 56921 * 56951 * 57041 * 57149 * 57203 * 57329 * 57413 * 57773 * 57839 * 57881 * 58013 * 58049 * 58193 * 58211 * 58451 * 58511 * 58601 * 58889 * 58979 * 59021 * 59063 * 59123 * 59369 * 59393 * 59399 * 59453 * 59513 * 59621 * 59723 * 59879 * 59981[/CODE]To do this, I took all the prime numbers p of the form 12 * n + 1 and 12 * n-1 for n from 1 to 10,000.
And for each of these prime numbers p, I looked for an odd exponent k such that 3^k == 1 (mod p), for k from 3 to 100,000.
Then, I took as exponent j noted above the lowest common multiple (lcm) of all these odd exponents.
Finally, I just had to enter this exponent for base 3 in Oliver's program :
[B]mono patf.exe -v 3 "j" 100000[/B]
Then I tested the abundance of the d given by the program.
Result : [U][I]abundance = 0.806[/I][/U]
I also tried this entry :
[B]mono patf.exe -v 3 "j" 1000000[/B]
Then I tested the abundance of the d given by the program.
Result : [U][I]abundance = 0.877[/I][/U]
I also tried this entry :
[B]mono patf.exe -v 3 "j" 10000000[/B]
It is impossible for me to test the abundance of the divisor d given by the program !

But now, for the first time, with this exponent j, I'm starting to believe that maybe we have abundance for base 3.
At least, I hope so, otherwise I'll have to look for an exponent k !!!

Who will dare to enter this instruction below into Oliver's program or Happy's new program to verify it ?
[B]mono patf.exe -v 3 "j" 1000000000[/B]