I have now completed testing of all unreserved bases on the project for all open sequences to >= 120 digits.

For this cycle, completed were bases in the 130 to 150 range. This included testing of bases 131, 137, 139, and 149.

Yoyo and others have already done this for sequences on bases < 130 or they are currently being worked on.

Mission accomplished. :-)

Jean-Luc, the only remaining sequences at < 120 digits besides a few stragglers with down-drivers that are being worked on by Yoyo are on your bases 77 and 88. Just a little nudge...:smile:

Unfortunately, I can't work on bases 77 and 88 at the moment.
I have big problems with my personal computer (not the 64C/128T one that I have at my disposal for a few days and that is not mine).
This computer is 12 years old !
In the next few days, I'm going to have to take it apart, and dust off all the components and I'm going to reinstall all the software hoping it will last me another 6-8 months.
Because I'm saving the money to own my own 64C/128T in a few months.
So the 77 and 88 bases are free the next week if you want to do some work on them.

I should also finish all the data analysis from last summer's big harvest.
It was long and complicated.
I will keep you posted in a few days about this.

[QUOTE=garambois;625164]Unfortunately, I can't work on bases 77 and 88 at the moment.
I have big problems with my personal computer (not the 64C/128T one that I have at my disposal for a few days and that is not mine).
This computer is 12 years old !
In the next few days, I'm going to have to take it apart, and dust off all the components and I'm going to reinstall all the software hoping it will last me another 6-8 months.
Because I'm saving the money to own my own 64C/128T in a few months.
So the 77 and 88 bases are free the next week if you want to do some work on them.

I should also finish all the data analysis from last summer's big harvest.
It was long and complicated.
I will keep you posted in a few days about this.[/QUOTE]

Wow! I will be curious to hear about your quest to purchase a 64C/128T machine. I'd like to get myself one at some point so I'd like to hear about your experience with researching and purchasing one.

OK, I am happy to finish off the 5 sequences each on bases 77 and 88 to >= 125 digits. That looks like how far you have been testing the bases so I will do that.

I'll start on them late Saturday or sometime Sunday. Let me know if that's OK. I won't make it a formal reservation since you still have the bases reserved.

[QUOTE=gd_barnes;625165]
I'll start on them late Saturday or sometime Sunday. Let me know if that's OK. I won't make it a formal reservation since you still have the bases reserved.[/QUOTE]
Perfect for me !

[QUOTE=gd_barnes;625165]Wow! I will be curious to hear about your quest to purchase a 64C/128T machine. I'd like to get myself one at some point so I'd like to hear about your experience with researching and purchasing one.
[/QUOTE]
There is a company in our area that builds such machines, very well ventilated.
The one I'm lent has two big GPUs that consume 300W each and throw the hot air into the tower.
The owner lends me 124 or 120 threads to run CADO-NFS for a few days, but he requires the GPUs to do something else in parallel.
But to run the GPUs, you need some CPU.
So when I run CADO-NFS, I have to run the "export OMP_DYNAMIC=true" instruction beforehand.
If I don't do this, the calculation gets stuck at the linear algebra step "lingen".
But according to the tests I made last summer with the same computer, it seems that the instruction "export OMP_DYNAMIC=true" slows down the calculations by a few percent, which is a pity.
But this computer is a real "bomb" as we say in French : 128T and 256 GB of ram.
Only weak point : the communication speed between the CPU and the ram.
But this problem should be solved with the arrival of DDR5.

With this future new computer, I have the ambition to relaunch the search for the "gaussian aliquot cycles" that we started with Andrew Walker, see here :
[URL="https://www.mersenneforum.org/showthread.php?t=21068"]https://www.mersenneforum.org/showthread.php?t=21068[/URL]
When I had started this work, I had talked to Paul Zimmermann about it and he had written a very powerful program in C language.
I intend to run this program on 64 Threads for 1 or 2 years !
I will then contact Chernykh to see if my future discoveries could interest him.
He seems to be interested only in Gaussian friendly pairs which remain in the set of integers and not in those which are in the set of complex numbers and which have a non zero imaginary part, [URL="https://sech.me/ap/"]see here[/URL].
But this could change, at least I hope so.
Otherwise, I will be obliged to create a special page on my site which would list these discoveries, especially since Paul Zimmermann's program has discovered a gaussian cycle of length 6, [URL="https://www.mersenneforum.org/showpost.php?p=528582&postcount=28"]see here[/URL].
All this is part of a more general framework of work that I explain [URL="http://www.aliquotes.com/etendre_sigma.html"]here[/URL], where the aim is to extend the sigma function to something other than integers.
I should also mention that I had also tried in some works to modify the iterative process always in order to try to generalize the problem of the aliquot sequences and to find a more general rule which would inform us about the behavior of the aliquot sequences : [URL="http://www.aliquotes.com/autres_processus_iteratifs.html"]see here[/URL].
These works were very fun and stimulating for the mind, but there was no general law of discovery !

In short, I have dozens of programs to run to continue all my work and my future 128-threaded computer will be saturated as soon as I get it !

:smile:

Thank you, Jean-Luc, for all of the info about your process to buy a new bigger machine in the future. I look forward to hearing more about your searches with such a beast of a machine!

Meanwhile:

I have completed bases 77 and 88 opposite-parities to >= 125 digits.

With that, all sequences* on the project have now reached >= 120 digits. I am officially done with the effort. :smile:

* There are still a few stragglers to go that are being worked on by Yoyo.

For your information, I will do a full update again before Sunday February 26th.

Thanks Gary for the 77 and 78 bases.
It looks like the dusting I did on my old machine worked.
I also recompiled all the programs : everything seems to work normally.
The machine does not crash anymore !

[B][SIZE="4"]Results of the 2022 Summer Harvest Data Analysis[/SIZE][/B]

I just finished analyzing the data from last summer's big harvest.
This analysis was very long and tedious.
I turned the data over and over.
There is no need for me to re-explain everything here.
To know the details of the analysis, you can read [URL="https://www.mersenneforum.org/showpost.php?p=612644&postcount=1845"]post #1845[/URL].

For those readers who are new to us and don't know what this "n^i project" is all about, here is a link to [URL="https://www.mersenneforum.org/showpost.php?p=586049&postcount=1328"]post #1328[/URL] which reminds us why this project was born.

The analysis of the summer 2021 data had been much faster.
The amount of data was much less, but more importantly, I looked for lots of new things for the 2022 data analysis compared to the previous analysis.

There were many "alerts" that led me to believe that I had stumbled upon something interesting.
Let me show you some examples of alerts :
1) The sequences 648^30 and 648^44 end with the prime number 53.
However, since the summer of 2022, Gary has managed to finish the calculation of the sequence 648^58 which also ends with the prime number 53 !
We notice that starting from 30, we always add 14 to the exponent.
The next exponent is 58 + 14 = 72.
Unfortunately, 648^72 is not a terminated sequence, I remain stuck at index 3 on a C171 cofactor.
Almost all such "alerts" remain unverifiable for the same reasons: the terms in the next sequence are far too large to verify.
I can also go backwards and look at 648^16, because 30 - 14 = 16.
And 648^16 does not end in 53.
So I think we are facing a false alert.
2) Another type of example of alert with amicable numbers (1184, 1210) :
1184^3 and 1210^12 end with the prime number 73.
1184^19 and 1210^3 end with the prime number 191.
No other powers of 1184 are present in the project data of all sequences that end in a prime <300.
So whenever 1184 is present, its amiable number is present for a given prime.
The converse is not true.
So I did all sorts of tests with the other pairs of amicable numbers.
Result : nothing !

And I spent weeks checking for such "alerts".
We are missing data with higher exponents for each base and no one will be able to bring us that data.
The 648^72 example is the "simplest" example and at the limit verifiable within 1 or 2 years, but the other examples not presented here are really out of reach.
Certainly if 648^72 ended in 53, there would be something !
But I bet that won't be the case !

I therefore consider that my analyses of this type have not given anything for this year : there is nothing serious !

BUT...

[SIZE="3"][B][U]A very surprising, counter-intuitive and inexplicable finding[/U][/B][/SIZE]

As usual, the surprise came not from what I was looking for but from something else entirely.
Please look at the data below :
[CODE]SAME PARITY
OPPOSITE PARITY

0 ---
1 [2, [2, 1]]

90 [3, [2, 2], [2, 4], [2, 164], [3, 1], [3, 5], [3, 247], [6, 152], [7, 77], [10, 124], [11, 15], [12, 2], [13, 15], [14, 76], [14, 80], [15, 1], [19, 15], [20, 2], [20, 8], [21, 21], [21, 55], [22, 80], [23, 3], [23, 109], [29, 69], [30, 82], [30, 92], [31, 79], [33, 1], [35, 49], [37, 11], [38, 30], [40, 14], [43, 15], [44, 58], [45, 1], [47, 13], [52, 34], [53, 27], [54, 10], [55, 53], [58, 18], [59, 49], [60, 86], [70, 70], [74, 6], [74, 20], [74, 78], [75, 45], [77, 5], [77, 19], [79, 97], [83, 43], [84, 30], [87, 1], [87, 17], [93, 49], [95, 5], [96, 42], [99, 3], [99, 71], [104, 58], [105, 1], [107, 11], [109, 13], [131, 57], [137, 3], [139, 33], [157, 5], [167, 63], [167, 79], [197, 23], [197, 27], [220, 16], [239, 3], [251, 33], [277, 9], [293, 11], [293, 23], [338, 2], [564, 2], [648, 56], [828, 24], [966, 6], [966, 34], [968, 8], [996, 12], [1250, 10], [2310, 36], [14288, 12], [223092870, 20]]
51 [3, [2, 55], [2, 305], [2, 317], [3, 2], [5, 38], [7, 4], [11, 2], [12, 1], [26, 1], [29, 2], [30, 1], [38, 11], [41, 2], [42, 1], [45, 34], [46, 1], [50, 73], [52, 1], [53, 2], [53, 30], [54, 1], [66, 1], [71, 2], [72, 1], [78, 1], [86, 1], [89, 2], [90, 1], [98, 45], [101, 2], [102, 1], [113, 2], [173, 2], [197, 2], [211, 2], [242, 41], [257, 2], [269, 2], [288, 1], [288, 3], [306, 17], [338, 13], [396, 11], [450, 1], [450, 19], [648, 1], [722, 37], [800, 21], [882, 5], [1058, 31], [1352, 1]]

0 ---
1 [5, [5, 1]]

51 [7, [2, 10], [2, 12], [2, 278], [3, 305], [7, 1], [7, 127], [11, 5], [13, 87], [14, 130], [17, 91], [18, 52], [18, 70], [18, 134], [21, 17], [23, 11], [26, 50], [26, 80], [30, 52], [30, 70], [35, 11], [40, 2], [41, 49], [42, 2], [45, 55], [56, 38], [68, 10], [70, 48], [75, 1], [75, 63], [79, 7], [80, 28], [92, 32], [93, 29], [94, 22], [97, 5], [99, 19], [102, 2], [193, 3], [200, 24], [200, 56], [211, 65], [271, 3], [277, 25], [293, 55], [385, 57], [450, 12], [648, 40], [968, 60], [8128, 38], [15015, 35], [131071, 25]]
78 [7, [2, 3], [2, 141], [2, 387], [2, 421], [3, 6], [3, 8], [3, 118], [3, 198], [7, 2], [7, 8], [10, 1], [12, 21], [13, 2], [14, 1], [14, 19], [14, 21], [15, 10], [17, 24], [18, 13], [19, 2], [20, 1], [21, 8], [22, 1], [22, 19], [26, 3], [26, 19], [28, 9], [28, 47], [34, 1], [34, 7], [34, 9], [37, 2], [38, 1], [38, 13], [43, 20], [45, 4], [46, 33], [47, 8], [47, 42], [48, 61], [50, 17], [53, 8], [58, 5], [58, 27], [61, 2], [62, 1], [66, 5], [66, 31], [72, 25], [74, 11], [76, 39], [77, 2], [77, 16], [78, 23], [84, 31], [96, 3], [96, 7], [105, 2], [151, 2], [163, 4], [220, 15], [223, 4], [227, 10], [229, 4], [242, 39], [271, 34], [338, 39], [450, 17], [496, 31], [564, 21], [696, 7], [722, 51], [800, 51], [888, 5], [14264, 7], [15472, 3], [19116, 15], [131071, 8]]

66 [11, [2, 60], [2, 316], [2, 480], [3, 15], [3, 189], [3, 303], [5, 15], [7, 143], [10, 20], [11, 1], [11, 137], [13, 31], [14, 14], [14, 28], [18, 76], [21, 1], [21, 47], [24, 2], [26, 36], [35, 83], [47, 47], [51, 1], [53, 15], [53, 67], [58, 2], [58, 24], [59, 81], [63, 85], [65, 33], [67, 3], [70, 2], [71, 61], [72, 4], [72, 58], [73, 3], [76, 28], [77, 15], [79, 3], [82, 2], [82, 54], [88, 54], [91, 1], [92, 30], [95, 63], [98, 2], [99, 13], [109, 15], [109, 75], [139, 5], [162, 70], [193, 57], [210, 58], [271, 23], [283, 27], [288, 4], [564, 30], [578, 50], [648, 42], [800, 54], [882, 50], [888, 26], [968, 32], [14288, 2], [14288, 4], [31704, 16], [223092870, 4]]
12 [11, [2, 499], [17, 2], [18, 1], [18, 55], [37, 50], [65, 2], [72, 63], [80, 3], [242, 25], [284, 19], [648, 41], [14288, 9]]

47 [13, [2, 358], [3, 3], [3, 31], [3, 67], [5, 9], [11, 3], [11, 27], [13, 1], [20, 72], [23, 49], [23, 77], [24, 70], [35, 1], [38, 86], [39, 91], [43, 3], [43, 17], [50, 62], [56, 8], [62, 2], [69, 1], [70, 34], [73, 11], [75, 13], [79, 23], [93, 1], [94, 12], [96, 6], [103, 5], [104, 24], [104, 64], [131, 9], [167, 17], [173, 69], [179, 25], [181, 23], [197, 3], [229, 21], [263, 7], [263, 15], [392, 48], [392, 54], [770, 28], [8191, 17], [15015, 9], [15015, 33], [6469693230, 2]]
5 [13, [50, 99], [271, 4], [648, 7], [968, 1], [1058, 7]]

27 [17, [3, 63], [6, 2], [7, 55], [13, 21], [15, 79], [17, 1], [19, 97], [24, 4], [38, 42], [39, 1], [45, 5], [46, 10], [51, 31], [55, 1], [58, 70], [60, 28], [78, 52], [84, 78], [85, 63], [92, 4], [96, 34], [99, 23], [119, 79], [131, 15], [157, 65], [271, 21], [396, 6]]
4 [17, [23, 2], [24, 1], [162, 39], [1152, 31]]

59 [19, [2, 76], [2, 190], [3, 275], [3, 327], [5, 233], [10, 2], [10, 12], [10, 44], [12, 4], [12, 150], [13, 3], [13, 141], [15, 21], [19, 1], [22, 32], [34, 96], [40, 6], [41, 7], [42, 10], [42, 22], [45, 15], [55, 29], [59, 85], [60, 30], [65, 1], [70, 28], [71, 21], [72, 74], [76, 66], [77, 1], [79, 13], [88, 62], [89, 9], [89, 19], [90, 2], [94, 70], [97, 29], [98, 6], [102, 54], [109, 27], [210, 12], [233, 41], [241, 51], [263, 21], [269, 11], [276, 10], [277, 11], [288, 16], [293, 17], [392, 6], [392, 26], [392, 42], [496, 30], [722, 30], [770, 14], [1058, 44], [8191, 15], [19116, 24], [131071, 21]]
7 [19, [2, 39], [2, 219], [2, 505], [41, 68], [193, 2], [338, 59], [1152, 27]]

44 [23, [3, 181], [7, 3], [7, 9], [11, 127], [17, 79], [18, 6], [18, 64], [21, 61], [23, 1], [40, 36], [44, 68], [47, 3], [54, 80], [55, 15], [57, 1], [57, 17], [65, 3], [69, 9], [71, 65], [74, 64], [77, 23], [77, 37], [79, 49], [85, 1], [86, 54], [88, 26], [93, 3], [99, 1], [101, 65], [105, 47], [127, 3], [137, 43], [173, 35], [200, 18], [242, 28], [263, 13], [293, 7], [496, 22], [564, 50], [800, 2], [8128, 8], [14536, 32], [19116, 8], [510510, 8]]
3 [23, [3, 12], [18, 17], [578, 1]]

21 [29, [5, 41], [14, 92], [18, 50], [18, 116], [22, 8], [26, 68], [29, 1], [46, 36], [47, 39], [48, 4], [61, 5], [63, 67], [83, 11], [88, 2], [88, 78], [91, 13], [97, 3], [229, 51], [283, 5], [306, 56], [82589933, 3]]
4 [29, [72, 43], [162, 11], [338, 57], [450, 21]]

39 [31, [2, 146], [3, 169], [5, 3], [5, 161], [11, 93], [11, 107], [12, 14], [19, 35], [19, 37], [31, 1], [31, 11], [40, 30], [44, 30], [56, 52], [61, 55], [70, 54], [70, 58], [80, 30], [87, 13], [91, 15], [92, 2], [92, 14], [95, 27], [97, 49], [103, 9], [113, 3], [119, 53], [179, 21], [210, 14], [220, 4], [227, 19], [239, 33], [269, 37], [281, 61], [882, 6], [1152, 4], [1152, 28], [1210, 10], [8191, 7]]
17 [31, [2, 5], [2, 101], [6, 17], [11, 38], [31, 2], [38, 19], [54, 67], [58, 1], [67, 2], [68, 1], [74, 21], [98, 3], [162, 27], [163, 8], [231, 2], [882, 1], [12496, 3]]

30 [37, [2, 68], [7, 79], [10, 74], [12, 98], [14, 2], [14, 138], [22, 74], [24, 6], [37, 1], [42, 16], [45, 11], [53, 87], [55, 43], [60, 94], [63, 3], [78, 4], [84, 32], [85, 81], [94, 16], [95, 37], [96, 48], [99, 5], [104, 4], [131, 45], [149, 43], [241, 47], [283, 33], [552, 18], [14316, 2], [15015, 27]]
43 [37, [2, 125], [2, 243], [3, 90], [5, 4], [7, 6], [10, 3], [10, 5], [11, 40], [14, 35], [22, 7], [22, 21], [22, 23], [23, 10], [26, 31], [28, 3], [28, 11], [29, 50], [35, 2], [44, 3], [44, 19], [47, 16], [51, 2], [67, 74], [71, 4], [74, 5], [75, 8], [77, 34], [79, 6], [83, 2], [83, 10], [84, 1], [88, 3], [90, 9], [96, 1], [97, 4], [131, 6], [139, 2], [220, 27], [239, 18], [241, 28], [251, 26], [277, 10], [293, 10]]

52 [41, [2, 6], [2, 8], [2, 112], [7, 11], [11, 57], [12, 32], [12, 56], [17, 65], [24, 18], [37, 73], [38, 54], [40, 20], [40, 52], [41, 1], [43, 25], [43, 81], [50, 16], [55, 21], [59, 39], [61, 9], [62, 40], [63, 1], [66, 78], [70, 38], [71, 5], [71, 7], [73, 27], [78, 64], [79, 17], [84, 2], [88, 22], [94, 46], [96, 50], [97, 7], [98, 10], [103, 21], [107, 3], [107, 61], [119, 29], [167, 13], [173, 9], [181, 5], [210, 4], [283, 17], [385, 11], [966, 2], [968, 2], [1210, 24], [1352, 4], [2310, 8], [9699690, 20], [223092870, 2]]
90 [41, [2, 23], [2, 47], [2, 117], [2, 281], [2, 373], [2, 405], [2, 411], [6, 5], [6, 13], [11, 14], [12, 119], [15, 2], [15, 6], [18, 39], [20, 3], [22, 67], [33, 2], [35, 8], [35, 34], [37, 20], [40, 11], [40, 21], [42, 5], [44, 49], [45, 8], [45, 22], [45, 36], [47, 2], [48, 1], [48, 3], [51, 18], [52, 25], [56, 1], [57, 2], [62, 3], [62, 13], [65, 4], [65, 26], [66, 25], [69, 2], [70, 13], [70, 23], [75, 2], [76, 1], [78, 15], [79, 2], [79, 4], [80, 1], [82, 11], [85, 8], [88, 1], [92, 1], [94, 11], [97, 48], [98, 41], [103, 2], [104, 1], [105, 10], [105, 30], [109, 2], [113, 16], [127, 26], [127, 44], [127, 52], [163, 2], [167, 20], [193, 32], [197, 10], [197, 20], [220, 17], [223, 2], [227, 2], [229, 2], [231, 4], [257, 28], [263, 8], [277, 14], [284, 11], [288, 57], [293, 4], [385, 42], [439, 2], [578, 3], [720, 3], [770, 13], [966, 31], [14288, 3], [14536, 5], [31704, 9], [8589869056, 3]]

97 [43, [2, 62], [2, 210], [2, 510], [6, 26], [6, 32], [7, 51], [7, 189], [10, 42], [11, 13], [13, 95], [17, 101], [17, 129], [18, 34], [19, 55], [20, 70], [22, 2], [22, 104], [24, 10], [24, 36], [26, 6], [26, 38], [26, 78], [26, 88], [28, 18], [29, 3], [29, 59], [33, 3], [33, 23], [33, 45], [34, 28], [34, 44], [37, 97], [38, 38], [43, 1], [43, 55], [47, 17], [51, 5], [51, 13], [60, 6], [62, 8], [62, 50], [66, 56], [67, 63], [68, 14], [68, 18], [69, 3], [69, 43], [70, 8], [70, 22], [75, 11], [75, 73], [79, 31], [80, 36], [80, 68], [82, 72], [83, 3], [88, 76], [93, 27], [95, 21], [102, 22], [102, 26], [104, 26], [104, 68], [104, 70], [105, 3], [107, 5], [109, 11], [127, 11], [137, 71], [157, 25], [167, 39], [179, 3], [179, 5], [191, 5], [191, 31], [199, 29], [227, 57], [231, 1], [241, 55], [257, 3], [263, 43], [269, 43], [276, 4], [277, 45], [439, 3], [496, 18], [496, 36], [552, 28], [660, 30], [828, 14], [1152, 12], [1155, 1], [1210, 28], [2310, 16], [19116, 4], [2147483647, 5], [200560490130, 10]]
121 [43, [2, 9], [2, 271], [3, 4], [3, 22], [3, 80], [3, 86], [6, 77], [7, 16], [7, 28], [11, 28], [11, 56], [13, 4], [13, 8], [13, 40], [13, 98], [14, 5], [14, 23], [14, 25], [15, 14], [17, 12], [17, 16], [19, 4], [20, 21], [21, 2], [21, 6], [21, 98], [23, 4], [24, 19], [24, 31], [28, 13], [31, 14], [31, 26], [34, 3], [35, 4], [39, 6], [40, 1], [40, 35], [42, 13], [43, 2], [44, 1], [45, 2], [46, 3], [46, 5], [48, 9], [50, 1], [51, 50], [52, 9], [53, 4], [54, 49], [57, 26], [58, 71], [59, 2], [59, 6], [59, 18], [60, 1], [62, 9], [63, 6], [65, 12], [65, 32], [66, 21], [67, 10], [70, 1], [70, 27], [71, 26], [73, 2], [73, 14], [74, 1], [75, 6], [75, 46], [78, 17], [79, 30], [79, 56], [80, 5], [82, 1], [86, 11], [87, 6], [87, 28], [90, 35], [91, 2], [91, 44], [92, 13], [92, 19], [93, 24], [94, 1], [96, 5], [97, 40], [98, 5], [98, 29], [99, 8], [104, 3], [105, 6], [107, 2], [119, 38], [131, 2], [139, 4], [157, 2], [181, 2], [191, 2], [200, 7], [223, 12], [227, 6], [227, 12], [227, 14], [231, 6], [233, 10], [271, 2], [271, 20], [277, 2], [306, 11], [385, 8], [396, 5], [648, 45], [722, 15], [770, 1], [968, 29], [1152, 9], [1152, 29], [1155, 2], [1250, 47], [15015, 12], [6469693230, 1]]

14 [47, [3, 9], [28, 34], [30, 10], [40, 22], [47, 1], [65, 47], [77, 13], [78, 80], [82, 26], [93, 17], [120, 44], [578, 22], [696, 2], [968, 40]]
3 [47, [2, 453], [98, 75], [200, 37]]

19 [53, [2, 20], [2, 78], [2, 214], [2, 450], [23, 101], [48, 50], [48, 70], [53, 1], [72, 18], [75, 31], [94, 2], [163, 61], [181, 53], [241, 3], [306, 8], [306, 50], [648, 30], [648, 44], [888, 14]]
1 [53, [2, 347]]

9 [59, [6, 128], [19, 25], [59, 1], [78, 44], [105, 19], [127, 25], [151, 27], [162, 62], [269, 47]]
108 [59, [3, 94], [7, 14], [7, 24], [7, 64], [10, 7], [10, 23], [11, 8], [11, 58], [11, 80], [13, 6], [13, 38], [14, 9], [14, 49], [15, 36], [17, 64], [19, 6], [20, 59], [20, 71], [24, 9], [24, 13], [26, 59], [26, 71], [28, 7], [28, 23], [29, 4], [29, 36], [31, 8], [31, 10], [33, 4], [33, 6], [35, 6], [38, 77], [39, 2], [39, 12], [42, 3], [44, 5], [45, 6], [45, 28], [45, 30], [51, 32], [51, 78], [52, 3], [52, 11], [55, 4], [56, 61], [57, 4], [58, 17], [60, 3], [60, 17], [61, 12], [65, 50], [66, 17], [69, 4], [69, 10], [69, 24], [70, 19], [70, 39], [71, 28], [73, 4], [73, 10], [74, 33], [75, 18], [75, 38], [77, 28], [84, 5], [86, 37], [86, 49], [87, 4], [87, 8], [89, 16], [90, 11], [95, 2], [97, 46], [99, 2], [103, 14], [103, 38], [104, 5], [119, 2], [131, 22], [137, 2], [149, 2], [151, 6], [167, 2], [173, 6], [179, 4], [179, 16], [191, 20], [193, 4], [193, 16], [197, 4], [199, 2], [199, 4], [200, 1], [211, 4], [220, 55], [233, 2], [251, 2], [263, 2], [271, 6], [281, 12], [385, 10], [439, 4], [564, 5], [720, 11], [14288, 7], [15472, 5], [131071, 4], [7420738134810, 1]]

15 [61, [30, 90], [39, 3], [43, 23], [45, 7], [52, 6], [52, 36], [61, 1], [67, 61], [75, 43], [76, 68], [104, 54], [137, 21], [284, 22], [496, 50], [1058, 12]]
1 [61, [60, 9]]

8 [67, [14, 6], [19, 53], [45, 47], [57, 13], [67, 1], [99, 55], [968, 4], [968, 58]]
1 [67, [2, 147]]

25 [71, [6, 14], [12, 58], [13, 53], [13, 113], [21, 11], [22, 36], [26, 66], [50, 2], [61, 11], [63, 33], [66, 70], [71, 1], [92, 60], [95, 25], [137, 33], [179, 23], [239, 45], [271, 7], [271, 53], [439, 31], [496, 8], [722, 12], [780, 30], [1210, 26], [510510, 4]]
5 [71, [2, 43], [2, 177], [18, 77], [162, 1], [722, 27]]

17 [73, [7, 133], [20, 4], [22, 94], [31, 3], [34, 2], [57, 47], [57, 81], [60, 62], [65, 57], [73, 1], [74, 74], [127, 19], [200, 2], [229, 3], [1210, 12], [131071, 3], [33550336, 18]]
30 [73, [2, 67], [3, 98], [5, 10], [6, 3], [10, 11], [10, 13], [12, 3], [17, 48], [23, 6], [28, 5], [34, 5], [43, 14], [46, 7], [63, 14], [73, 12], [76, 9], [78, 3], [86, 7], [97, 2], [98, 1], [105, 12], [137, 4], [173, 14], [263, 10], [496, 43], [968, 11], [1184, 3], [2310, 25], [19116, 5], [30030, 11]]

11 [79, [3, 113], [6, 4], [17, 39], [34, 56], [50, 6], [79, 1], [96, 26], [151, 67], [181, 45], [211, 53], [1250, 2]]
1 [79, [82, 5]]

10 [83, [2, 376], [5, 5], [5, 57], [56, 44], [60, 88], [78, 12], [83, 1], [84, 46], [95, 57], [120, 28]]
1 [83, [5, 36]]

13 [89, [2, 32], [2, 82], [3, 111], [7, 101], [37, 23], [47, 31], [57, 5], [68, 12], [89, 1], [113, 73], [450, 28], [660, 44], [510510, 2]]
---

11 [97, [7, 43], [38, 34], [43, 57], [68, 2], [97, 1], [98, 80], [103, 43], [107, 67], [131, 17], [293, 57], [392, 18]]
4 [97, [2, 51], [2, 73], [19, 38], [1250, 9]]

9 [101, [31, 29], [45, 93], [55, 45], [73, 43], [73, 79], [99, 69], [101, 1], [1210, 2], [7420738134810, 8]]
4 [101, [2, 301], [2, 425], [29, 12], [181, 16]]

8 [103, [2, 50], [6, 16], [24, 90], [44, 74], [50, 12], [72, 62], [103, 1], [396, 10]]
1 [103, [242, 47]]

20 [107, [5, 223], [7, 165], [7, 171], [14, 94], [41, 29], [76, 78], [85, 31], [101, 31], [105, 61], [107, 1], [119, 3], [220, 2], [227, 41], [229, 17], [229, 39], [231, 57], [251, 3], [293, 9], [496, 60], [82589933, 13]]
1 [107, [2, 255]]

14 [109, [3, 249], [10, 80], [18, 10], [28, 98], [30, 46], [55, 49], [79, 33], [83, 7], [109, 1], [131, 21], [276, 6], [338, 4], [1352, 16], [14288, 6]]
3 [109, [85, 4], [578, 11], [660, 5]]

7 [113, [5, 31], [55, 3], [58, 50], [102, 12], [113, 1], [239, 31], [578, 28]]
9 [113, [2, 11], [3, 38], [19, 50], [20, 5], [21, 70], [34, 15], [38, 3], [72, 21], [73, 22]]

8 [127, [5, 53], [53, 39], [97, 73], [101, 75], [127, 1], [139, 29], [211, 9], [1250, 6]]
2 [127, [2, 7], [127, 2]]

8 [131, [3, 141], [19, 3], [30, 2], [33, 9], [66, 2], [66, 88], [119, 49], [131, 1]]
1 [131, [162, 37]]

8 [137, [2, 396], [42, 42], [54, 28], [82, 16], [97, 57], [137, 1], [191, 47], [968, 54]]
0 ---

4 [139, [61, 27], [77, 49], [139, 1], [284, 4]]
0 ---


3 [149, [3, 55], [59, 45], [149, 1]]
0 ---

6 [151, [96, 70], [103, 63], [151, 1], [162, 16], [210, 40], [15472, 38]]
1 [151, [18, 79]]

5 [157, [20, 18], [20, 34], [60, 12], [65, 83], [157, 1]]
2 [157, [241, 2], [242, 1]]

8 [163, [3, 47], [35, 9], [163, 1], [163, 51], [220, 26], [284, 32], [780, 4], [800, 8]]
3 [163, [281, 2], [293, 2]]

3 [167, [45, 35], [109, 3], [167, 1]]
0 ---

4 [173, [7, 31], [57, 59], [173, 1], [277, 3]]
0 ---

11 [179, [14, 40], [40, 72], [60, 36], [61, 31], [87, 25], [109, 29], [179, 1], [200, 52], [227, 13], [241, 5], [14316, 8]]
0 ---

3 [181, [12, 6], [149, 63], [181, 1]]
0 ---

12 [191, [7, 63], [18, 112], [40, 84], [54, 22], [56, 2], [78, 14], [89, 15], [191, 1], [306, 34], [385, 1], [439, 13], [1152, 2]]
10 [191, [12, 5], [47, 48], [68, 25], [94, 3], [220, 3], [284, 5], [450, 3], [578, 5], [1184, 19], [1210, 3]]

9 [193, [23, 39], [26, 2], [51, 67], [63, 5], [98, 4], [193, 1], [242, 10], [1155, 19], [15472, 40]]
3 [193, [37, 10], [62, 55], [882, 31]]

2 [197, [15, 85], [197, 1]]
0 ---

4 [199, [2, 110], [23, 71], [199, 1], [450, 36]]
0 ---

8 [211, [2, 46], [3, 211], [33, 21], [43, 21], [48, 58], [65, 63], [76, 14], [211, 1]]
1 [211, [338, 1]]

4 [223, [69, 19], [105, 23], [162, 68], [223, 1]]
0 ---

2 [227, [58, 76], [227, 1]]
0 ---

3 [229, [23, 33], [229, 1], [241, 49]]
0 ---

4 [233, [14, 4], [120, 72], [229, 63], [233, 1]]
0 ---

6 [239, [5, 93], [6, 204], [19, 7], [20, 14], [181, 3], [239, 1]]
1 [239, [1152, 1]]

7 [241, [2, 346], [19, 11], [33, 89], [57, 7], [59, 25], [241, 1], [269, 35]]
1 [241, [2, 279]]

3 [251, [191, 3], [251, 1], [263, 3]]
0 ---

13 [257, [20, 112], [33, 39], [46, 2], [46, 56], [47, 19], [54, 4], [101, 15], [104, 8], [104, 40], [193, 13], [257, 1], [288, 44], [2147483647, 9]]
4 [257, [13, 10], [18, 3], [93, 2], [96, 9]]

3 [263, [29, 7], [263, 1], [648, 8]]
0 ---

5 [269, [53, 71], [101, 7], [127, 7], [269, 1], [1155, 41]]
0 ---

1 [271, [271, 1]]
1 [271, [18, 7]]

3 [277, [13, 83], [79, 61], [277, 1]]
4 [277, [2, 103], [71, 6], [720, 1], [510510, 5]]

12 [281, [6, 40], [14, 46], [19, 27], [34, 4], [42, 48], [50, 38], [93, 21], [96, 28], [181, 7], [197, 5], [281, 1], [828, 4]]
1 [281, [37, 74]]

3 [283, [2, 86], [48, 10], [283, 1]]
1 [283, [392, 55]]

3 [293, [15, 47], [281, 67], [293, 1]]
1 [293, [2, 53]][/CODE]
How to read this data ?
For example, if we take the two lines below :
[CODE]5 [157, [20, 18], [20, 34], [60, 12], [65, 83], [157, 1]]
2 [157, [241, 2], [242, 1]][/CODE]
This means that there are 5 sequences of the same parity that end in the prime 157 (which are 20^18, 20^34, 60^12, 65^83 and 157^1) and 2 sequences of opposite parity that end in the prime 157 (which are 241^2 and 242^1).
[U]Now, given the rarity of sequences of opposite parity which end, there should be for each prime number p many more sequences of the same parity which end with this prime number than of sequences of opposite parity.
This is not the case ![/U]
You can see that the prime numbers that are exceptions are [B]2, 5, 7, 37, 41, 43, 59, 73, 113, and 277.[/B]
I remind you that the work only analyzed prime numbers <300.
There may be other primes >300 for which this will also be the case.
I am checking with Karsten to see if he can modify his scripts to make such phenomena directly apparent.
The case of the prime number 59 is particularly spectacular, because 108 sequences of opposite parity end in 59 while there are only 9 sequences of the same parity which end in 59 !
I see there a real paradox and I cannot explain it, because I noticed that the bases which are the doubles of squares do not explain this paradox !
[B]If someone finds an explanation, I would be very interested ![/B]

[SIZE="3"][B][U]What will we do now ?[/U][/B][/SIZE]

Again, I want to thank all the people who have worked so hard on this project so far.
I would like to remind you that we are not at all sure that we will reach the goal we have set for ourselves.
The laws linking the base, the exponent and the prime number or the cycle on which the sequences end may not exist.
So to all, I say : continue this project only if it pleases you, because I can't guarantee any result.
It's no problem for me if someone wants to withdraw.
But I hope, of course, that you will all continue the work for this project !

If some of you want to continue to do calculations for this project, here are some possible directions.

1) It seems difficult to add exponents to the bases we have started so far.
The calculations would become too long and so this option is to be discarded for the moment.
We have done it for base 2 and it is very difficult : I see that even with a 128 threads machine, it is very long.
On the other hand, it is very important that we continue as far as possible to complete as many high exponent sequences of the same parity as possible to add high "y" coordinate points on the curves I am looking for.

2) The other option is to add bases to add points with high "x" coordinates on the possible curves I am looking for.
We now have all the bases for prime numbers up to 500 and that should be enough.
If some of you want to add bases, I recommend adding bases that will follow the bases 106, (107), 108, (109), 110, 111, 112 and (113) that I am currently working on, in order to have all the bases exhaustively beyond 113.
It would also be interesting to add 2 or 3 couples of amicable numbers and to continue to initialize bases that are part of the C28 cycle that we started a long time ago, and also some bases which would be factorial numbers.

[B]And also, if anyone has their own ideas of things they would like to look at, they can suggest the bases that interest them.[/B]

On my side, I will continue to look for new ideas to look at the data in other different ways.

[B]Do not hesitate if you have any comments, or even criticisms to make following this post.
[/B]
:smile:

[QUOTE=garambois;625591]
We notice that starting from 30, we always add 14 to the exponent.
The next exponent is 58 + 14 = 72.
Unfortunately, 648^72 is not a terminated sequence, I remain stuck at index 3 on a C171 cofactor.
Almost all such "alerts" remain unverifiable for the same reasons: the terms in the next sequence are far too large to verify.
:smile:[/QUOTE]

I can crack the C171 for you, if you like. While I doubt we'll see the pattern continue, satisfying such curiosity is as good a reason as any to run a factoring job!

This is great if you can afford to crack it.
But at the next indexes, you may come across a C200, and there, it will be another level of difficulty !
But I'm excited if you go for it...
For the ECM methods, I went up to t40 and it still runs on my PC with one thread.
Let me know if you take over !

:smile:

I ran up through t50 on the c171 today.

I "might" attempt t55 tomorrow, but I'm not sure my cluster will actually do t55. I know it didn't like t60 when I tried it on something a while ago.