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[QUOTE=EdH;627718]
The merge is interesting to me in particular, because it shows the merging of the two base 116 sequences only a couple terms prior to the merge with 6552. This makes me wonder if there may be several sequences within the project which have merged with each other, but gone unnoticed because they don't merge with the main project. If 6552 wasn't involved, would the two merging have been caught? I may have to set up a program to see if I can find any such merges within the whole set. [/QUOTE] If there had not been a merger with 6552, I would not have noticed. [B]It might indeed be interesting to have a list of mergers of the project sequences between them.[/B] But I don't expect to find anything but randomness in these mergers ! At least that's what I assume. [QUOTE=EdH;627718] Bummer on the computer issues cropping up again. I can only suggest checking the things you've already done in the past: heat issues and power supply failure. If it is the power supply and you keep restarting, it may damage the motherboard. As to when to upgrade, that will be quite a decision to make - sorry I can't be of more help. [/QUOTE] Thanks Edwin, but this computer is from 2011 ! There's nothing left to do. I didn't even dare to run the calculations of the programs like for example the program that creates the file "regina", for fear of compromising it. I still don't know what I'm going to do. I'm even going to have trouble finishing the initializations of the bases for which I committed myself ! But I will try... And if my future computer isn't up to snuff, I won't be able to restart the dozens of programs I've put on hold and therefore won't be able to do my work. [QUOTE=EdH;627718] More directly to the project, a lot of the matched parity sequences have current terms above 150 digits, which take me a while to bring down to 135* for that sub-project. Your addition of base 116 has helped that thread and the others you currently have pending will help, too. In light of the above, I'm wondering if I should initialize some new bases rather than bring existing sequences that have a current term >150 down to 135. Do you think it would be better to add more bases or extend existing sequences in regards to your research efforts? [/QUOTE] [B]I really think it would be very interesting to initialize new bases, as it would provide more data for a smaller effort. We should initialize factorial bases, and especially compound bases up to 200 in an exhaustive way, but without necessarily terminating all sequences of the same parity ![/B] Thus, there would be a continuity in the curves. [B]Anyone who wants to initialize new bases <200 can do so if they feel like it ![/B] I think we have enough bases that are prime numbers, as we are past 500 now. Anyway, if we manage to initialize all the bases to 200 within 1 or 2 years (by the summer of 2024), we will then mark a long pause of several years in the calculations. That should be enough and if there will be a discovery, it will be more due to the quality of the data analysis than to the quantity of the data. [QUOTE=EdH;627718] A last note: As you already know, I will from time to time, at least, want to move my machines to other areas, which may include team projects that take me away for extended periods. I should be able to still maintain the threads, but I might not be able to work on sequences. [/QUOTE] Edwin, there's really no problem if you feel like working on something else. I also want to work on something else quite often. I'm planning to go back to a lot of the old and equally interesting work that I was doing before I started this project. Besides, my experience shows me that sometimes you have to let your mind rest a bit and then come back to it later : then you see things differently. Let's go slowly, according to our desires, let's enjoy ourselves. It really doesn't matter if all the bases <200 are not initialized by summer 2024 : I'll do with what I have. The work we've done so far is colossal ! And as said above, I will quietly continue the data analysis. Maybe also some of us will feel like joining again the main project : I hope it will be extended to 5,000,000 soon ! [B]But if people, especially newcomers to the project, want to continue the calculations at a very fast pace, I will gladly continue to make the updates and of course I will use the data in my analyses ! Please don't see this message as a drop in motivation on my part, it's quite the opposite ![/B] :tu: [QUOTE=EdH;627718] * You might want to change the 145 to 135 on the main project page where you mention the matched parity thread.[/QUOTE] I will do this in the next update. Thanks Edwin. |
Thanks Jean-Luc!
A large quantity of my "farm" is 2011 vintage, with some Core2 Quads and AMDs that might even be older. But, although they are still running factoring programs, they are power hungry. If you would like, I can take the four initializations that you have listed off your hands and turn them toward the matched parity sub-project. I'm thinking of handling them (and future bases) in this manner for now: - all matched parity with a term below 136 added to the available list for matched parity to be terminated by whomever - all matched parity with a term 136 through 145, I'll bring down to <136 - all opposite parity taken to >99 term with fully ECM'd cofactor* Does that sound workable? * Gary took everything to 120 digits, but for now, to keep up with other things, I'd like to work with your original value of 100 digits for the open opposite parity lower bound. This can be readdressed later. |
Thank you very much Edwin, but I will try to finish these 4 initializations for these 4 bases myself.
Otherwise, the parameters you suggest for the other initializations you might undertake seem excellent. |
Sounds good, Jean-Luc. If I decide to initialize any new base, I'll mention it [STRIKE]here[/STRIKE] in the updates thread prior to running it.
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OK, many thanks Edwin !
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[QUOTE=garambois;627735]If there had not been a merger with 6552, I would not have noticed.
[B]It might indeed be interesting to have a list of mergers of the project sequences between them.[/B] But I don't expect to find anything but randomness in these mergers ! At least that's what I assume. . . .[/QUOTE]I did a full search of the current set of sequences in all the tables and found all merges traced back to the main project sequences. There were none that included only sequences within this project. |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=628315&postcount=44"]See here.[/URL] |
[QUOTE=EdH;628303]I did a full search of the current set of sequences in all the tables and found all merges traced back to the main project sequences. There were none that included only sequences within this project.[/QUOTE]
OK, thanks Edwin. Great job ! This is remarkable and quite curious ! I would have bet that mergers between sequences only of the project would have been found. |
Based on your experiences, what would be around a core week or two of work to finish a sequence (on average)?
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Base 180 can be added at the next update.
Initializing base 182. |
taking base 118 up to 120 digits
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I'm back from vacation.
For your information, I will update during coming week. |
[QUOTE=kruoli;628492]Based on your experiences, what would be around a core week or two of work to finish a sequence (on average)?[/QUOTE]
Sorry, but I'm not sure I understand the question. You are asking about the average time to terminate a sequence, is that correct ? Because it varies very much depending on the sequence, but I think you already know that. That makes me assume that I'm misunderstanding the question... |
[QUOTE=RichD;628769]Base 180 can be added at the next update.
Initializing base 182.[/QUOTE] Thank you very much Rich ! Please, can you put this type of ads on the [URL="https://www.mersenneforum.org/showthread.php?t=28467"]new thread provided for this purpose[/URL], being careful to respect the codes ? :smile: |
[QUOTE=birtwistlecaleb;629015]taking base 118 up to 120 digits[/QUOTE]
Excellent, thank you very much ! Once the work is finished, you will just have to ask for an update of the complete 118 base as announced in the previous post in the thread provided for this purpose. You can request it like this : "Update 118". |
[QUOTE=garambois;629173]Sorry, but I'm not sure I understand the question.
You are asking about the average time to terminate a sequence, is that correct ? Because it varies very much depending on the sequence, but I think you already know that. That makes me assume that I'm misunderstanding the question...[/QUOTE] Thank you for your answer! Going from the time when I wrote that, I think I was already tired. You are correct, I [I]should[/I] know that but had likely a brain fart. Where I wanted to go with this: Maybe newcomers to the project might be afraid a work unit takes too long. But since I do not even know how many potential candidates are there that would like to participate, this worry might be taking it too far. |
To make it very simple :
Sequences of the same parity fast enough if there is no prime 3 in the terms decompositions. Of course, if we start with 130 digits numbers, it is not the same thing as if we start with 170 digits numbers ! It is up to each one to know if he can decompose numbers of 130 or 170 digits and in how much time. Opposite parity sequences : Same remark as before. And it's not the same if the sequence keeps yo-yoing, or if it grows very fast if its terms are very "brittle" (I don't know if "brittle" is the right word in English to designate a term which has lots of small prime numbers in the decomposition of its terms ? Perhaps "friable" or "frangible" would be more appropriate ?) And it's not at all the same thing to compute a sequence up to 120 digits or up to 170 digits ! In short : you can compute a sequence in minutes or hours, or in days, weeks or even months ! |
[QUOTE=garambois;629259]To make it very simple :
Sequences of the same parity fast enough if there is no prime 3 in the terms decompositions. Of course, if we start with 130 digits numbers, it is not the same thing as if we start with 170 digits numbers ! It is up to each one to know if he can decompose numbers of 130 or 170 digits and in how much time. Opposite parity sequences : Same remark as before. And it's not the same if the sequence keeps yo-yoing, [B]or if it grows very fast if its terms are very "brittle" (I don't know if "brittle" is the right word in English to designate a term which has lots of small prime numbers in the decomposition of its terms ?[/B] Perhaps "friable" or "frangible" would be more appropriate ?) And it's not at all the same thing to compute a sequence up to 120 digits or up to 170 digits ! In short : you can compute a sequence in minutes or hours, or in days, weeks or even months ![/QUOTE] You can just say it has many small factors, but if you want to describe it like that I think abundant might be a good choice. |
[QUOTE=garambois;629259]Opposite parity sequences : Same remark as before.
And it's not the same if the sequence keeps yo-yoing, or if it grows very fast if its terms are very "brittle" (I don't know if "brittle" is the right word in English to designate a term which has lots of small prime numbers in the decomposition of its terms ? Perhaps "friable" or "frangible" would be more appropriate ?) And it's not at all the same thing to compute a sequence up to 120 digits or up to 170 digits ![/QUOTE] [QUOTE=birtwistlecaleb;629294]You can just say it has many small factors, but if you want to describe it like that I think abundant might be a good choice.[/QUOTE] "Abundant" already has a definition ([i]n[/i] is "abundant" if [i]n[/i]'s aliquot sum is greater than [i]n[/i]), so don't use it like that. [i]n[/i] can be called "[I]m[/I]-smooth" if [I]all[/I] of its factors are less than [I]m[/I], but I don't think there's a particular term for an [I]n[/I] where just [I]some[/I] factors are less than [I]m[/I]. |
Although still not quite right, perhaps "complex" would describe a term with many factors rather than few.
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"Abundant" would actually be a good choice to describe the overall concept you're trying to talk about (not a number with a lot of small factors, but a number with a large aliquot sum relative to the number itself, causing the sequence to accelerate upward). I think that's the more important point, rather than the exact factor form.
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Yes, I can say "abundant", that will make the concept and the message clear.
But in French we have the word "friable" which brings a nuance to the word "abundant". The most friable numbers possible are powers of 2. |
[QUOTE=garambois;629350]Yes, I can say "abundant", that will make the concept and the message clear.
But in French we have the word "friable" which brings a nuance to the word "abundant". The most friable numbers possible are powers of 2.[/QUOTE] Wikipedia describes "friable" as synonymous with "smooth", as I defined that term a few posts ago. Powers of 2 are 2-smooth (or 2-friable), but a full term in an aliquot sequence isn't smooth or friable generally, as it usually has at least one large factor to go along with the small ones, and [I]m[/I]-smooth/[I]m[/I]-friable numbers need [I]all[/I] of their prime factors to be less than or equal to [I]m[/I]. |
Yes Happy, sorry, you are right.
I was using the word "friable" wrong even in French, I just checked. "Abundant" is more appropriate ! |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=629413&postcount=71"]See here[/URL]. |
I think I should've said this earlier, but you didn't add my name code for bases 118 and 129, can you add these soon?
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[QUOTE=birtwistlecaleb;629694]I think I should've said this earlier, but you didn't add my name code for bases 118 and 129, can you add these soon?[/QUOTE]
Yes, you are right : sorry, this will be done in the next update. |
In response to [URL="https://www.mersenneforum.org/showpost.php?p=629954&postcount=1524"]this post from another thread[/URL] and the following posts :
Honestly, if you want to finish sequences for base 13 up to exponent 160 and have the computing power, don't go without. This data will be important and I will take it into account in the next big harvest. There is nothing "stupid" about doing this and this data is just as important as other data, but it is harder to get. It is also not complicated to add exponents to base 13 on the project page. But in this case, I will also extend base 12 to exponent 160 to harmonize everything. Ideally, it would even be very interesting to extend the other small bases. Especially base 2, which we are doing, but also the other bases smaller than 13. But the calculations become so long for such large exponents, that I think it is better to continue to add the bases up to 200 in an exhaustive way, that is what will give the most data with the least effort. On the other hand, I sometimes have some remorse and think that this project is engaging people to use their computers to calculate sequences that start on powers of integers at the expense of the sequences of the main project. I hope that other people than me will benefit from all these calculations that we do in this project ! [B]Should I extend bases 12 and 13 to exponent 160 on the project page ?[/B] If there is no advice to the contrary, I will do so in the next update. If someone thinks that it is better not to extend these two bases, I thank them for presenting their arguments here : maybe they will be right and maybe we won't do it then ? |
Sure, extending 13 to 160 is fine with me. I'll add the same-parity powers 153, 157, 159 to my queue to terminate.
I terminated 147,149,151,155 last July (as Ed noted in the other thread). |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=630223&postcount=87"]See here[/URL]. |
By the way, I'm only reserving the sequences that are less than 120 digits. As my goal is to get all of the sequence to 120 digits, I only need to work on sequences less than 120 digits. When this page gets updated again, please consider this.
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[QUOTE=garambois;630224]Page updated
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=630223&postcount=87"]See here[/URL].[/QUOTE]Thank you (and Karsten) for all the work at your end, Jean-Luc! |
[QUOTE=birtwistlecaleb;630225]By the way, I'm only reserving the sequences that are less than 120 digits. As my goal is to get all of the sequence to 120 digits, I only need to work on sequences less than 120 digits. When this page gets updated again, please consider this.[/QUOTE]
OK, sorry, I'll fix that in the next update. [QUOTE=EdH;630237]Thank you (and Karsten) for all the work at your end, Jean-Luc![/QUOTE] You're welcome Edwin ! Thanks to all of you for all your hard work ! As for me, I don't have much time at the moment, but I am following your progress attentively, while waiting to be able to examine all these new data... |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=630782&postcount=94"]See here[/URL]. |
My severe disability to get aliqueit to run gnfs properly makes it take a lot more effort than it should to do aliquot sequences. To prevent the tediousness of grabbing the numbers themselves, I'm going to unreserve the two sequences and find something else to do. Hopefully someone else does the same as I did with a working version of aliqueit.
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I've figured out the issue of why ggnfs doesn't work, and have finally made aliqueit work with ggnfs. I'm gonna try this project again, as I now have a working copy of aliqueit.
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Okay, got it.
Don't forget to specify which sequences you're going to work on in the appropriate thread. Many thanks ! |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=631465&postcount=106"]See here[/URL]. |
Page updated again.
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=632164&postcount=110"]See here[/URL]. |
Page updated again.
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=633486&postcount=131"]See here[/URL]. |
Thanks Jean-Luc (and Karsten)! My lists in the other threads were getting pretty long.:smile:
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Page updated again.
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=634152&postcount=138"]See here[/URL]. |
If anyone is interested in downdriver runs, [URL="http://www.factordb.com/sequences.php?se=1&aq=143%5E70&action=all&fr=0&to=100"]143^70[/URL] has had the downdriver since index 1. It's now at index 42 and term size 143. It's mixed parity and I have other interests in mind for now.
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[QUOTE=EdH;634371]If anyone is interested in downdriver runs, [URL="http://www.factordb.com/sequences.php?se=1&aq=143%5E70&action=all&fr=0&to=100"]143^70[/URL] has had the downdriver since index 1. It's now at index 42 and term size 143. It's mixed parity and I have other interests in mind for now.[/QUOTE]
On it! Taking 143^70. |
Page updated again.
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=634571&postcount=142"]See here[/URL]. I'm going to take some time off until mid-August. After that, I'll do another update. Then I'll take some time to examine the data. This summer, I'll be looking mainly at the 0, 1 and 2 indexes of the project sequences. I'll resume the classic examination of prime numbers and cycles that end sequences in a few months or in the summer of 2024, when we've exhaustively initialized all bases < 200. By then, I hope to have the computer of my dreams at my disposal (64C/128T) to restart the execution of dozens of pending programs. And also to calculate same-parity sequences with terms of more than 160 digits ! :smile: |
[QUOTE=VBCurtis;634381]On it!
Taking 143^70.[/QUOTE] This ran down to 28 digits, then picked up a fast-rising driver. I got it to 138 digits or so before my summer-heat shutdown. I'll continue it in a few weeks when I power its host machine back on. |
[QUOTE=VBCurtis;635028]This ran down to 28 digits, then picked up a fast-rising driver. I got it to 138 digits or so before my summer-heat shutdown. I'll continue it in a few weeks when I power its host machine back on.[/QUOTE]Hmm. . . The db still shows it at index 42. Unless Jean-Luc needs it to be 140+, it can probably be left at 138. The only reason I moved it was to take it off the index 1 listing and then, since I noticed the dd, I let it run down a little before offering it to someone else to experience for a while.
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Sigh, that means I forgot to update factorDB before shutting the machine down. I'll fix it around 15 Aug.
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[QUOTE=VBCurtis;635071]Sigh, that means I forgot to update factorDB before shutting the machine down. I'll fix it around 15 Aug.[/QUOTE]That sounds good. At a term of 143 digits and these few posts, I don't foresee anyone working it.
I'm having temperature issues with some of my machines now and it's only just touching 80F here. I cleaned and redid the heat sink on one of them with little improvement.:sad: One of my Z620s has an issue, but not as bad. The second CPU has a convex top and only the near-center portion of the heat sink appears to be in good contact. (I wonder if a vice might fix it. . .) |
[QUOTE=EdH;635051]Unless Jean-Luc needs it to be 140+, it can probably be left at 138.[/QUOTE]
This is of course no problem if the sequence remains at 138 digits instead of 140. It's wonderful to have taken the calculations this far ! :smile: |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=636088&postcount=156"]See here[/URL]. |
[SIZE="3"][B][U]Work in progress for this summer 2023 :[/U][/B][/SIZE]
I'm now going to look at the project sequences in a different way to other years. I'll resume the usual work when we have all the sequences <200 exhaustively. In the meantime, I'm going to look at numbers of iterations at the end of which the sequences are terminated. I've never tried to approach the problem from this angle before. We'll see if this new way of looking at things yields anything interesting. It will mainly concern the small <10 indexes of the sequences, and particularly indexes 1 and 2. Of course, I'll be commenting on this work in the near future to keep you up to date, even if nothing interesting comes of it. But one thing is certain : for bases 2, 3 and other bases that are small primes, the exponents of sequences that end after one iteration are listed in the OEIS. This shows that looking at sequences from this angle is not without interest ! |
Welcome back! And, thanks for the updates.
Your list with items of focus is quite welcome. I've been wondering which direction to work. Currently, I'm slowly running index 1 composites at the c158 level. Would you prefer the current status, with possibly one/two c158s completed to index 2 per day, or would you like more of the smaller index 1 composites completed from new bases? If I initialize new bases in our drive toward exhausting <200, I can add many more completed index 1 sequences, but those will, obviously, be smaller terms. I'm looking at whether I should go ahead and terminate a fair number of the additions to the "Easier" thread, so the initialization would also contribute to the overall number of terminated sequences in the "big picture," as well as possibly adding other items of interest such as cycles. |
[QUOTE=EdH;636108]
Would you prefer the current status, with possibly one/two c158s completed to index 2 per day, or would you like more of the smaller index 1 composites completed from new bases? If I initialize new bases in our drive toward exhausting <200, I can add many more completed index 1 sequences, but those will, obviously, be smaller terms. I'm looking at whether I should go ahead and terminate a fair number of the additions to the "Easier" thread, so the initialization would also contribute to the overall number of terminated sequences in the "big picture," as well as possibly adding other items of interest such as cycles.[/QUOTE] Thank you Edwin for your interest in this work ! During this month of August, I actually think it will be more profitable for work in progress to initialize bases <200 to have more sequences with indexes>=2, and even beyond, if possible. For example, for the new 152 base added today, I'll be able to scan up to exponent 75 for index 2 but only up to exponent 46 for index 3, because the 152^47 sequence has only been calculated up to index 2. For the same base 152, I'll be able to scan up to exponent 46 if I want to work up to index 10, because the 152^47 sequence has only been calculated up to index 2.. I'm thinking of working up to index 10 for all bases <153, as we have all bases < 153 exhaustively. This number will change again if more bases are initialized between now and August 20 at 18:00 GMT when I will proceed with the big scan. I'm going to search for sequences of integers (A list of exponents here, for a given base : exponents of sequences stopping on a prime at the index 1, ditto for index 2, ditto for index 3...) and the number of known integers in a sequence in the OEIS must be greater than 3 or 4. Indeed, it's better to calculate "easy" sequences in this month of August. I'll explain all this in more detail in my review in a while. |
Thanks Jean-Luc,
I will move toward more initializations and note them in the update thread. I'll work toward making sure the smallest sequences include index 1 completion. I'm not sure how high the lower bound will actually be for these, but we'll see. This will create a pretty large number of candidates for the "Easier" thread, but I will also work on those after you start the gathering operation. |
[SIZE="3"][B][U]A few more explanations about this summer's work[/U][/B][/SIZE]
I'm currently examining the data in greater detail as part of my preparatory work. Before automating the work with a program, I try things out "by hand" to develop a kind of intuition so that I can then adjust the program parameters correctly. To help you understand what I'm going to do, let's take the examples of base 2 and base 3 below. [B]Base 2[/B], I look at [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/2.html"]this table[/URL] and extract the data "by hand". List of exponents whose sequences end at index 1 : 2,3,5,7,13,17,19,31... This sequence is listed in the OEIS and is the famous Mersenne prime numbers ([URL="https://oeis.org/A000043"]https://oeis.org/A000043[/URL]). List of exponents whose sequences end at index 2 : 6,11,18,27,41... This sequence is not listed in the OEIS. List of exponents whose sequences end at index 3 : 25,29,98,131,199... This sequence is not listed in the OEIS. ... ... [B]Base 3[/B], I look at [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/3.html"]this table[/URL] and extract the data "by hand". List of exponents whose sequences end at index 1 : 3,7,13,71,103... This sequence is listed in the OEIS ([URL="https://oeis.org/A028491"]https://oeis.org/A028491[/URL]). List of exponents whose sequences end at index 2 : 2,17,89,191... This sequence is not listed in the OEIS. List of exponents whose sequences end at index 3 : 4,11,23,25,27,35... This sequence is not listed in the OEIS. ... ... But for base 3, a lot of Open-End sequences remain. I must therefore make sure that all Open-End sequences have been calculated up to the index in question, to ensure that an exponent with a different parity does not result in an uncalculated sequence ending at that index. [B][U]A few other preliminary observations[/U][/B] For example, for [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/23.html"]base 23[/URL], I only note one sequence that ends at index 1, but there are several that end at index 5, and I find this curious ! For [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/31.html"]base 31[/URL], I find that the sequences of exponents 7,17,31 end at index 1. The OEIS tells me that the next exponent will be 5581 ([URL="https://oeis.org/A128002"]https://oeis.org/A128002[/URL]) ! I note dozens of little findings like this that tickle my fancy ! And I want to try to see a little more clearly, hence this summer's work. [B][U]One difficulty : limited data for bases >19[/U][/B] For example, for [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/101.html"]base 101[/URL], I have a single sequence ending at index 1, 3 ending at index 2 and 2 ending at index 3. That's very little ! And curiously enough, the situation seems even worse for bases that are not prime numbers, such as [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/102.html"]base 102[/URL], for which there are very few exponents whose sequences end at the index 1, or 2, or 3. And it may be that this summer's work can't be done for all the bases for which we have less than 140 exponents, i.e. up to base 19. [B]After this observation, I don't know whether the additions of bases >153 will add anything to this summer's analysis. But we're not safe from a surprise, you never know ![/B] But even if the initialization of sequences >153 should not be useful for this summer's work-in-progress, it will be useful later on anyway. In a while, I'll let you know what I find, eventually ! |
[QUOTE=garambois;636150]For [URL="http://www.aliquotes.com/aliquotes_puissances_entieres/all/31.html"]base 31[/URL], I find that the sequences of exponents 7,17,31 end at index 1.
The OEIS tells me that the next exponent will be 5581 ([URL="https://oeis.org/A128002"]https://oeis.org/A128002[/URL]) ![/QUOTE] Confirmed: [url]http://factordb.com/sequences.php?se=1&aq=31%5E5581&action=last20&fr=0&to=100[/url]. (This took less than a second to generate, so it wasn't too hard on the database. The prime had already been proven via N-1.) |
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Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=636475&postcount=164"]See here[/URL]. |
Thanks for all the updates!
I might try to initialize more <200 bases, but the db is the bottleneck while trying to take the mixed parity to 100 digits. I will give priority to the index 1 side of any new initializations, since that is a focus for this summer. I believe the lower bound for the index 1 listing is at a cofactor size of 151 digits. |
I have started the process of ensuring all index 1 sequences with < 151 digit cofactors, are moved to index 2, at least. The following bases are NOT being initialized, but should be completed in regards to index 1 work as described above:
164 165 166 168 I think this should be done within a short time. |
Thank you so much Edwin for all your hard work !
[QUOTE=EdH;636505] I might try to initialize more <200 bases, but the db is the bottleneck while trying to take the mixed parity to 100 digits.[/QUOTE] I've also encountered the same problem. I almost solved it by doing the following : Every time I calculate a new iteration for a sequence, I add the row directly to FactorDB in real time. And I only do this for terms > 50 digits. In fact, FactorDB completes all terms < 50 digits on its own. This prevents FactorDB from becoming overloaded, as calculations go too fast when the terms in a sequence are < 50 digits. And rarely, FactorDB is crashed, so calculations can't be added to it. Only then do I have to enter the data manually, which is a pain. So I still have to store the calculations on the hard disk in case I need to do this. But as mentioned above, this is very rare ! |
I'm already doing something similar to what you describe, but I only send the largest factor for each line and then after all lines have been sent, I use Aliqueit to send the entire elf, which only causes one page and a very few "leftover" IDs (sometimes none) to be added. I limit the process to 1800 lines at a time and then it pauses for an hour. This fits in with other things I run at the same time. But there is an inefficiency having to do with the delay that I've been looking at putting some work into. I plan to adjust the delay based on time to reset.
BTW, I hope to have all the index 1 <151 digits finished for the four bases listed by tomorrow morning. There are six left currently. If I can, I'll add the 17x bases, other than what RichD is working on starting tomorrow. By next week I'll look at finishing inialization for all of them. |
[QUOTE=EdH;636591]
BTW, I hope to have all the index 1 <151 digits finished for the four bases listed by tomorrow morning. There are six left currently. If I can, I'll add the 17x bases, other than what RichD is working on starting tomorrow. By next week I'll look at finishing inialization for all of them.[/QUOTE] I hope that for all these sequences, we'll have many that end of a small index and even <=2 ! As for me, I finally have the program I wanted for this summer. I'm doing the final tests before the big scan on Sunday. I've run into a few problems with taking into account OpenEnd sequences (which limit the size of the end index I can examine so that the results are definitive and not provisional) and with sequences that end with cycles. Incidentally, this last problem caused me to spot errors with the cycles on the project page. I hadn't quite taken Karsten's explanations into account when new cycles were added to the project. This has now been rectified. |
I "think" all the 17x bases with terms < 151 digits have been taken past index 1. The new listings for the ones that are left are posted.
I ran the ones from RichD's initialization of 177 after he posted the update. @RichD: I hope I didn't tread on any of your work when I ran some of the base 176 sequence index 1s. I was finishing one or two of them on Friday. I didn't catch your initialization post until the last one finished. I'm thinking they were probably outside the exponents you would work, anyway. I'm totally away from 176 now. |
Having read all the above information and [URL="https://www.mersenneforum.org/showpost.php?p=636765&postcount=168"]this one too[/URL], I'm thinking I'll wait another 3 or 4 days before carrying out my scan.
This will give me all the bases, right down to base 170. Furthermore, as Rich has worked his way down, he's reached base 176, so all that's missing are bases 170, 171, 172, 174 and 175 to have all the bases exhaustively down to 200. Perhaps I'll scan down to base 200. Only these 5 bases will be missing. I've already done some preliminary work up to base 28. I've managed to present the data in an "easy-to-read" way by importing the data into a LibreOffice spreadsheet. I'm going to work out a few more details to make it as presentable as possible, so that you too can try to look at the tables and find something. [B]Let's postpone the scan until Thursday August 24 at 6:00 pm GMT.[/B] Thank you all for your hard work. It's only by working together that we can be as effective as we are. :smile: |
[QUOTE=EdH;636763]@RichD: I hope I didn't tread on any of your work when I ran some of the base 176 sequence index 1s. I was finishing one or two of them on Friday. I didn't catch your initialization post until the last one finished. I'm thinking they were probably outside the exponents you would work, anyway. I'm totally away from 176 now.[/QUOTE]
No worries. I've "touched" every exponent through 90 to get it into the database. I only have about a dozen that still needs movement to C100 or termination. It is on a box I am getting frustrated with. It keeps crashing at least once a day. After I complete this initialization I am going to open the box, check the fans and replace the thermal paste. I've also "touched" some of the low non-parity exponents in the lower 170s. This causes FDB to add (up to) several hundred terms until it reaches a C50. This is costly to one's limits. I do this on occasion to help with whoever gets the base will have a jump start. |
I had noticed the 170s had been "touched" as you mentioned. Thanks! I have been doing that via calling for elfs, and that doesn't seem to go against my limits.
Right now I'm bumping right up against the limits while initializing, even with all the attempts to minimize uploads, because I'm doing several things at once. |
As you can imagine, I didn't wait for the big scan to start the analysis.
I'm taking a very close look at the sequences ending with a prime number at index 1. I'll take care of the following indexes after the big scan, although I've already had a good look at things up to base 28. I've written a tiny program of just a few lines that gives me the exponents of those sequences that end at index 1 for all bases up to 28 up to exponent 10000. This is possible, because if n<28, the decomposition of n^10000 into prime factors is very fast, since obviously, n^1, n^2, n^3 ... n^9999, n^10000 are all n-friable in the worst case (if n is prime). After that, I don't have to do any factorization of large numbers, just a primality test on the term of the index sequence 1. Of course, I only perform a pseudo-primality test ("is_pseudoprime" instruction on Sage software). This can be done in a "reasonable" time (several days up to base 28). [U]I made an observation :[/U] The project's sequence analysis didn't show me any sequences that end at index 1 for bases (<=28) 6, 10, 12, 14, 15, 20, 22, 24, 26. These aren't all bases that aren't prime numbers, because for bases 18, 21 and 28, there are : 18^2, 18^14, 21^1, 21^9, 21^13 and 28^2 which end at index 1 with a prime number. The little program I mentioned above has added two sequences at the time of writing (its execution is not yet complete) : 18^3194 and especially 12^4414. Our project had not given any sequences for base 12 ! But what confuses me is that when I entered 12^4414 and 18^3194 in FactorDB, the prime numbers of index 1 of these two sequences were not known as prime numbers, because for a few minutes, FactorDB printed me the mention "Not all factors known". Only after several minutes did the message change to "Prime". I deduce that, for the moment, nobody has really been interested in these sequences which end at index 1 for bases that are not primes. To corroborate this, OEIS doesn't seem to know the sequence starting with 2, 14, 3194. No one seems to have done any research on this, except for bases that are p primes, which seem to interest mathematicians. OEIS mentions them like this : "Numbers n such that (p^n - 1)/(p-1) is prime". This is because if base B is a prime number, then sigma(B^i)-B^i = (B^i-1)/(B-1). This equality does not hold if B is not a prime number. [U]So I naturally ask myself the following questions :[/U] [B]If i is any natural integer and B is a composite number (not prime), is it possible to demonstrate other than by calculation that sigma(B^i)-B^i can never be prime or on the contrary that it will be prime for certain values of i (an infinity of i) ? In fact, I don't even know if this existence or this infinity of values of i is demonstrated in the case of B prime ? Should I try with my computer to find a number i such that the sequences 6^i, 10^i, 14^i... end at index 1, which I haven't found yet ?[/B] Because my intuition tells me that if certain bases which are not prime numbers have such exponents i, there is no reason why 6, 10, 14 and others should not have them. But that's just my intuition ! |
[QUOTE=garambois;636840]
In fact, I don't even know if this existence or this infinity of values of i is demonstrated in the case of B prime ? Should I try with my computer to find a number i such that the sequences 6^i, 10^i, 14^i... end at index 1, which I haven't found yet ? Because my intuition tells me that if certain bases which are not prime numbers have such exponents i, there is no reason why 6, 10, 14 and others should not have them. But that's just my intuition ![/QUOTE] I believe that you'd be wasting your time for 6^i. If my maths is correct, for odd i, the aliquot is always even, and for even i, it is divisible by 5. So it will never be prime. |
Page updated.
Many thanks to all for your work ! [URL="https://www.mersenneforum.org/showpost.php?p=636948&postcount=179"]See here[/URL]. |
[QUOTE=Brownfox;636912]I believe that you'd be wasting your time for 6^i. If my maths is correct, for odd i, the aliquot is always even, and for even i, it is divisible by 5. So it will never be prime.[/QUOTE]
Thank you very much Brownfox for your interest in this question. Unfortunately I can't reproduce your calculations. In the end, unless I am mistaken, I find this : [CODE]s(6^i) = sigma(6^i) - 6^i = sigma(2^i * 3^i) - 2^i * 3^i = (2^(i + 1)) / (2 - 1) * (3^(i + 1)) / (3 - 1) - 2^i * 3^i = (2^(i + 2) * 3^i - 2^(i + 1) - 3^(i + 1) + 1) / 2 [/CODE] And unfortunately, to continue with i = 2j or i = 2j + 1, I don't know how to go on to establish that the quantity is divisible by 5 or by 2 ! It doesn't seem elementary to me, but maybe it is - I'm just a mathematical amateur ! I think that's how you arrived at your conclusions, isn't it ? But I understand the process if you did like this. I have no idea how difficult and time-consuming it would be for redo this work for 10, 14, 15 ... [B]Do you think the process can be automatically programmed to get out the bases for which there's no need to search for exponents (that don't exist) whose sequences would end at index 1 ?[/B] |
Sorry, I'm stupid : of course s(6^i) is even if i is odd !
I even have a demonstration of this in the general case on my site. However, I still can't establish that s(6^i) is divisible by 5 if i is even. As for the slightly more general case n = p*q with p and q prime, the demonstration that s(n^i) may or may not be prime as a function of p, q and i just seems abominable ! |
I am very much also an amateur.
I'm sure there are more elegant ways of proving it, but: Working modulo 5, 2^4 == 3^4 == 1 So 2^(i+4) == 2^4.2^i == 1.2^i == 2^i and similarly for 3^(i+4) So we only need to consider i=1..4 - higher values we can just work out by subtracting 4 repeatedly until it is in the range 1..4. For i = 1 and 3, we've agreed that the aliquot will be even. So it just leaves us with i=2 and i=4. The aliquot of 6^2 is (1+2+4)(1+3+9) - 36 = 91 - 36 = 55 == 0 (mod 5) So i=2,6,10 etc will all ==0 (mod 5) The aliquot of 6^4 is (1+2+4+8+16)(1+3+9+27+81) - 1296 = 3751 - 1296 = 2455 == 0 (mod 5) So i=4,8,12 etc will all ==0 (mod 5) This proof would also work for any prime == 3 (mod 5), so you can also exclude 2*13 = 26^i, 2*23 = 46^i etc. Not sure how we would code a general proof though. Beyond my (non-existent) pay grade :smile: |
Thank you Brownfox for these explanations.
It inspired me to write a little program. For all bases of the form b = 2 * q (with q prime), I test all even exponents "e" from 2 to 1000. Then, I try to divide all s(b^e) by 3, 5, 7, 11 ... 997 (successive primes up to 997), until I find a prime that divides them all. And if I find one, then I know that most probably, s(b^e) can never be prime, so don't bother looking. If I don't find one, then it seems reasonable to try to find odd exponents for which s(b^e) will be prime. I'm well aware that this doesn't demonstrate anything, but it's already an empirical method that gives me some indications. Here's what my program displays for all bases of the form b = 2 * q for q < 100 : [CODE]For base 6 = 2 * 3 for all even exponents, seems divisible by 5 For base 10 = 2 * 5 for all even exponents seems divisible by 3 For base 22 = 2 * 11 for all even exponents seems divisible by 3 For base 26 = 2 * 13 for all even exponents seems divisible by 5 For base 34 = 2 * 17 for all even exponents seems divisible by 3 For base 46 = 2 * 23 for all even exponents seems divisible by 3 For base 58 = 2 * 29 for all even exponents seems divisible by 3 For base 82 = 2 * 41 for all even exponents seems divisible by 3 For base 86 = 2 * 43 for all even exponents seems divisible by 5 For base 94 = 2 * 47 for all even exponents seems divisible by 3 For base 106 = 2 * 53 for all even exponents seems divisible by 3 For base 118 = 2 * 59 for all even exponents seems divisible by 3 For base 142 = 2 * 71 for all even exponents seems divisible by 3 For base 146 = 2 * 73 for all even exponents seems divisible by 5 For base 166 = 2 * 83 for all even exponents seems divisible by 3 For base 178 = 2 * 89 for all even exponents seems divisible by 3[/CODE] I see the results mentioned in the previous post for b = 6, b = 26. For b = 46, it seems that all s(46^e) with even e are also divisible by 3 in addition to 5. But I don't show the 5, the 3 is enough to prevent s(n^e) from being prime. I've also tried running my program for bases of the form b = 3 * q with odd exponents, here's the result : [CODE]For base 15 = 3 * 5 for all odd exponents seems divisible by 3 For base 33 = 3 * 11 for all odd exponents seems divisible by 3 For base 51 = 3 * 17 for all odd exponents seems divisible by 3 For base 69 = 3 * 23 for all odd exponents seems divisible by 3 For base 87 = 3 * 29 for all odd exponents seems divisible by 3 For base 123 = 3 * 41 for all odd exponents seems divisible by 3 For base 141 = 3 * 47 for all odd exponents seems divisible by 3 For base 159 = 3 * 53 for all odd exponents seems divisible by 3 For base 177 = 3 * 59 for all odd exponents seems divisible by 3 For base 213 = 3 * 71 for all odd exponents seems divisible by 3 For base 249 = 3 * 83 for all odd exponents seems divisible by 3 For base 267 = 3 * 89 for all odd exponents seems divisible by 3[/CODE] Here, the prime numbers q are prime numbers of the form 5 modulo 6. For bases of the form b = 5 * q with odd exponents : [CODE]For base 95 = 5 * 19 for all odd exponents seems divisible by 5 For base 145 = 5 * 29 for all odd exponents seems divisible by 5 For base 295 = 5 * 59 for all odd exponents seems divisible by 5 For base 395 = 5 * 79 for all odd exponents seems divisible by 5 For base 445 = 5 * 89 for all odd exponents seems divisible by 5[/CODE] Here, the prime numbers q are prime numbers of the form 9 modulo 10. For bases of the form b = 7 * q with odd exponents : [CODE]For base 91 = 7 * 13 for all odd exponents seems divisible by 7 For base 287 = 7 * 41 for all odd exponents seems divisible by 7 For base 581 = 7 * 83 for all odd exponents seems divisible by 7 For base 679 = 7 * 97 for all odd exponents seems divisible by 7[/CODE] Here, the prime numbers q are prime numbers of the form 13 modulo 14. [B]I don't have to be very clever to notice that for all bases of the form b = p * q (with 2 < p < q), the s((p * q)^e) all seem divisible by p if q is a prime such that q == p - 1 modulo 2p.[/B] Of course, this is not demonstrated by me but just observed with the computer. But I guess I could have deduced that from [URL="https://www.mersenneforum.org/showpost.php?p=554258&postcount=450"]warachwe's post 3 years ago[/URL] ! This result seems to be well known. Perhaps as a first step, I should try to program a sort of sieve to remove all bases that have two prime numbers in their prime decomposition, p and q, such that p divides q + 1. This would already remove a series for which s(b^e) cannot be a prime. |
I know it's a totally different format (it can be changed, if desired), but here is a listing of all the index 1 terminations I found up through exponent 10000 for everything up through base 300. This includes bases not in the tables, but excludes any squares, etc. that I noticed, like 225 and 289. I thought you may like to compare:[code]2^2 = 3
2^3 = 7 2^5 = 31 2^7 = 127 2^13 = 8191 2^17 = 131071 2^19 = 524287 2^31 = 2147483647 2^61 = 2305843009213693951 2^89 = 61897001964269013744...11 <27> 2^107 = 16225927682921336339...27 <33> 2^127 = 17014118346046923173...27 <39> 2^521 = 68647976601306097149...51 <157> 2^607 = 53113799281676709868...27 <183> 2^1279 = 10407932194664399081...87 <386> 2^2203 = 14759799152141802350...07 <664> 2^2281 = 44608755718375842957...51 <687> 2^3217 = 25911708601320262777...71 <969> 2^4253 = 19079700752443907380...91 <1281> 2^4423 = 28554254222827961390...07 <1332> 2^9689 = 47822027880546120295...11 <2917> 2^9941 = 34608828249085121524...51 <2993> - - - - - 3^3 = 13 3^7 = 1093 3^13 = 797161 3^71 = 37547332574898624019...73 <34> 3^103 = 69575965298821529689...13 <49> 3^541 = 66308439547181843673...01 <258> 3^1091 = 17308478920290520561...73 <521> 3^1367 = 83892899771563540051...93 <652> 3^1627 = 94460759517675329357...93 <776> 3^4177 = 43097389851648611435...81 <1993> 3^9011 = 10929399054294972442...73 <4300> 3^9551 = 48314093098582972867...73 <4557> - - - - - 5^3 = 31 5^7 = 19531 5^11 = 12207031 5^13 = 305175781 5^47 = 17763568394002504646...31 <33> 5^127 = 14693679385278593849...31 <89> 5^149 = 35032461608120426773...81 <104> 5^181 = 81566305849981556583...81 <126> 5^619 = 11491393399729176129...31 <433> 5^929 = 55090160396265551027...81 <649> 5^3407 = 61481546263270361105...31 <2381> - - - - - 7^5 = 2801 7^13 = 16148168401 7^131 = 85053461164796801949...57 <110> 7^149 = 13850221271010340870...01 <126> 7^1699 = 11051436530648345552...57 <1436> - - - - - 11^17 = 50544702849929377 11^19 = 6115909044841454629 11^73 = 10511531995000535984...53 <76> 11^139 = 56700023252179573962...49 <144> 11^907 = 34927334101562340034...17 <944> 11^1907 = 86268172235636277197...17 <1985> 11^2029 = 96773887190642445536...89 <2112> 11^4801 = 53245320390468147223...01 <4999> 11^5153 = 19792771928746725123...33 <5366> - - - - - 12^4414 = 64128347646873409132...19 <4764> - - - - - 13^5 = 30941 13^7 = 5229043 13^137 = 33967064818628105570...61 <152> 13^283 = 14682075627206248721...83 <315> 13^883 = 34103491244661234780...83 <983> 13^991 = 68973281719462518324...03 <1103> 13^1021 = 18070969763323218316...01 <1137> 13^1193 = 71653606768823597575...21 <1328> 13^3671 = 16101453136591379562...03 <4089> - - - - - 17^3 = 307 17^5 = 88741 17^7 = 25646167 17^11 = 2141993519227 17^47 = 42362279579873318721...67 <57> 17^71 = 14379819517246113852...27 <87> 17^419 = 22593217065142772428...47 <515> 17^4799 = 52511413852558392920...47 <5904> - - - - - 18^2 = 523 18^14 = 749592419975463223 18^3194 = 43793665374710595259...23 <4010> - - - - - 19^19 = 109912203092239643840221 19^31 = 24327031889148383810...01 <39> 19^47 = 70169234660105574400...41 <59> 19^59 = 15530661393266602867...21 <75> 19^61 = 56065687629692436349...01 <77> 19^107 = 37270292131592212274...41 <136> 19^337 = 48382801987590421041...41 <430> 19^1061 = 31790554703354833065...01 <1356> 19^9511 = 93374041609950189489...01 <12161> - - - - - 21^1 = 11 21^9 = 595686490411 21^13 = 115854226785351611 21^1975 = 18120424307138201442...99 <2612> - - - - - 23^5 = 292561 23^3181 = 20597935370569888091...01 <4331> - - - - - 43^5 = 3500201 43^13 = 40911050578149780601 43^6277 = 45524625586914719910...01 <10252> - - - - - 47^127 = 49393536963321376486...97 <211> - - - - - 48^2 = 4339 - - - - - 50^1 = 43 50^265 = 25301275063753255738...53 <451> - - - - - 52^2 = 2969 - - - - - 53^11 = 178250690949465223 53^31 = 54519091977165126485...23 <52> 53^41 = 95347060825111462805...01 <69> 53^1571 = 13224759997125890740...23 <2708> - - - - - 55^1 = 17 55^957 = 12621911005228186777...53 <1666> 55^3223 = 57940135654309620411...29 <5609> - - - - - 57^1 = 23 57^439 = 39353053338088543392...07 <771> - - - - - 59^3 = 3541 59^13 = 1809873235795386729241 59^479 = 29832220681487207334...61 <847> - - - - - 60^2 = 8893 60^4 = 35330011 60^3962 = 29825430008008357257...93 <7046> - - - - - 61^7 = 52379047267 61^37 = 19013087418896543607...97 <65> 61^107 = 17870680721834899593...67 <190> 61^769 = 13819485447752210574...29 <1372> - - - - - 63^1 = 41 63^49 = 11034386366700170193...41 <89> 63^4069 = 24646598480277448042...41 <7322> - - - - - 65^1 = 19 - - - - - 66^26 = 46759531962894123742...91 <48> 66^4066 = 42400380040854572945...31 <7399> - - - - - 67^19 = 75141059740006460252...97 <33> 67^367 = 22382503220919718540...17 <669> 67^1487 = 35783900748120262555...17 <2714> 67^3347 = 11293295814640160058...17 <6111> 67^4451 = 10949716774026896304...77 <8127> - - - - - 70^4 = 43805011 - - - - - 71^3 = 5113 71^31 = 34987327501952148741...81 <56> 71^41 = 11389227221780038891...41 <75> 71^157 = 63454541278097423322...77 <289> 71^1583 = 49758580484399335862...93 <2929> - - - - - 73^5 = 28792661 73^7 = 153436090543 - - - - - 74^2 = 4373 74^62 = 82384919023674818736...73 <116> 74^2638 = 11396139221100949862...73 <4932> - - - - - 77^1 = 19 - - - - - 79^5 = 39449441 79^109 = 88973023266894547658...81 <205> 79^149 = 71505920148918592801...81 <281> 79^659 = 44071573620102802852...81 <1249> - - - - - 83^5 = 48037081 83^2713 = 35079287510764448085...41 <5205> - - - - - 85^1 = 23 - - - - - 89^3 = 8011 89^7 = 502628805631 89^43 = 75733427719941306155...11 <82> 89^47 = 47516849737606379366...31 <90> 89^71 = 28987302297978853232...51 <137> 89^109 = 34596172362033561979...41 <211> 89^571 = 14361874325794263253...51 <1112> - - - - - 90^38 = 50182009986227537532...57 <75> 90^146 = 57377163147556475228...57 <286> 90^248 = 12344543510943497076...71 <486> 90^2546 = 87249254806827911356...57 <4976> - - - - - 93^387 = 34935243919651907743...83 <762> - - - - - 97^17 = 62065212901958868055...41 <32> 97^37 = 33750711668629399004...41 <72> 97^1693 = 41905871767788717448...81 <3362> - - - - - 98^1 = 73 98^7 = 114958810847863 98^34 = 67084982393600703484...83 <68> 98^267 = 60576049407220719080...63 <532> 98^5739 = 59072742777040759319...63 <11428> - - - - - 101^3 = 10303 101^337 = 28595847000743753034...37 <674> 101^677 = 84250027412281653446...77 <1355> 101^1181 = 12692361624275050333...81 <2366> 101^6599 = 32865904703785758935...99 <13225> - - - - - 102^38 = 46425286083666733022...93 <77> - - - - - 103^19 = 17191235814481380275...33 <37> 103^313 = 10220006239818029046...61 <629> 103^1549 = 75207455698135649563...41 <3116> - - - - - 107^17 = 29800143499669677234...01 <33> - - - - - 109^17 = 40070679725506875803...21 <33> 109^1193 = 41341695861183700749...41 <2429> - - - - - 111^1 = 41 111^37 = 25744373390093691827...41 <76> - - - - - 113^23 = 14845204255764341964...83 <46> 113^37 = 82164532059904939917...61 <74> 113^6563 = 20165386903140422691...83 <13473> - - - - - 115^1 = 29 - - - - - 117^3 = 999727 - - - - - 119^7 = 80962692493561 - - - - - 120^2 = 36781 120^74 = 19895043880102318056...01 <155> 120^398 = 89839767801710390151...41 <828> 120^1658 = 52703400908292422292...41 <3448> - - - - - 122^2 = 11597 122^4 = 214841299 - - - - - 126^34 = 64645536290288713525...03 <72> 126^114 = 69212119997472952473...03 <240> - - - - - 127^5 = 262209281 127^23 = 19369349555573971915...57 <47> 127^31 = 13108252682696435092...97 <64> 127^167 = 17173185241060704934...77 <350> 127^5281 = 12220242474824934224...01 <11109> 127^8969 = 82262652490103319308...61 <18867> - - - - - 129^1 = 47 129^7 = 318325481092991 - - - - - 130^2 = 22811 - - - - - 131^3 = 17293 131^31 = 33225139497115729176...81 <64> 131^263 = 53506606151164072352...53 <555> - - - - - 132^26 = 31379791047463962809...07 <56> 132^1208 = 10352297228422567260...83 <2563> - - - - - 137^11 = 2346320474383711003267 137^19 = 29117351391180423666...27 <39> 137^1009 = 65692131330287553956...41 <2154> 137^2939 = 48776200171595552168...27 <6278> - - - - - 139^163 = 14843398770909926592...61 <348> 139^173 = 39965141738510587838...61 <369> 139^3821 = 20877220805114163861...01 <8187> - - - - - 140^722 = 61233396509214319480...77 <1550> - - - - - 143^3 = 560113 143^5 = 11461057561 143^15 = 40981434096161388991...13 <32> 143^317 = 33424926919813438946...69 <683> - - - - - 148^2 = 21713 - - - - - 149^7 = 11016462577051 149^13 = 12054793463967560892...01 <27> 149^17 = 59416196556663663338...01 <35> 149^317 = 53676646036760066287...01 <687> 149^3251 = 72160061207047840658...51 <7063> - - - - - 150^2 = 48571 - - - - - 151^13 = 14145198807859836896...13 <27> 151^29 = 10333327940128053377...29 <62> 151^127 = 35808080994033043170...77 <275> 151^4831 = 28942838031928861035...81 <10525> 151^5051 = 68622971321693039315...01 <11004> - - - - - 155^1 = 37 - - - - - 156^1042 = 38727013192002641660...13 <2286> - - - - - 157^17 = 13714168861685806957...01 <36> 157^107 = 58632472621049055847...07 <233> 157^2791 = 36543090698468377165...07 <6127> - - - - - 161^1 = 31 - - - - - 162^2 = 42643 162^14 = 17148270217591009042...63 <32> 162^2522 = 49876580643299627563...43 <5573> - - - - - 163^7 = 18871143464293 163^43 = 82139484858291195128...33 <93> 163^241 = 84663778481653663558...01 <531> 163^1637 = 13854580957225227391...61 <3620> 163^2543 = 24185940755892646091...33 <5624> - - - - - 167^3 = 28057 167^19 = 10268449314768133715...97 <41> 167^373 = 71307572147279967850...21 <827> 167^1213 = 86093708193671374184...21 <2694> 167^2203 = 26563044515708299916...57 <4895> - - - - - 171^1 = 89 171^2043 = 59837070765276954292...09 <4562> - - - - - 172^622 = 33029147357096999872...99 <1391> - - - - - 173^3 = 30103 173^2687 = 24794231975932584239...43 <6012> - - - - - 179^19 = 35792133474583554973...81 <41> - - - - - 181^17 = 13343300528800077827...97 <37> 181^19 = 43713986862401934971...99 <41> 181^157 = 15858559988918209013...37 <353> - - - - - 191^17 = 31536510869370512992...57 <37> 191^1399 = 77077421678964486767...89 <3189> - - - - - 193^5 = 1394714501 193^317 = 17312736804876068150...01 <723> - - - - - 197^31 = 68579914550147596482...87 <69> 197^47 = 35290398157683760358...47 <106> 197^283 = 11007430001537119966...07 <648> - - - - - 199^577 = 13853512411712860641...01 <1325> 199^1831 = 80041960217482415634...01 <4207> - - - - - 201^1 = 71 201^337 = 78586240002479464957...71 <776> 201^5307 = 60221544556820735514...99 <12223> - - - - - 203^1 = 37 203^129 = 96771540236517231853...17 <297> 203^143 = 19529547653842738970...73 <330> - - - - - 205^1 = 47 205^3 = 2405339 205^7 = 4279218807420443 205^397 = 16420083839328666062...83 <918> 205^1915 = 28690390070248586875...71 <4427> - - - - - 209^1 = 31 209^327 = 78516521537613622294...71 <758> - - - - - 211^41 = 94050557115859600625...41 <93> - - - - - 215^57 = 24876013933597572739...89 <133> - - - - - 221^1 = 31 - - - - - 223^239 = 79160639510756862755...13 <559> 223^241 = 39365794422304280279...01 <564> 223^449 = 11028857723012922280...21 <1053> - - - - - 227^5 = 2666986681 227^1061 = 24509025448292101662...01 <2498> 227^2687 = 19380716946652303838...77 <6329> - - - - - 229^11 = 398341412240537151131351 229^29 = 11947970478340699127...81 <67> - - - - - 233^113 = 13987209862978737945...41 <266> 233^9511 = 36031047947621303383...63 <22514> - - - - - 235^1 = 53 235^2657 = 23445657104671761207...93 <6300> - - - - - 237^1 = 83 237^3 = 6662347 237^99 = 65380264847867173336...27 <235> 237^1011 = 38542354069605814552...87 <2401> 237^2199 = 61231274158616445448...27 <5222> - - - - - 239^5 = 3276517921 239^109 = 73925495864171640450...41 <257> 239^2549 = 14443503790843019025...41 <6061> - - - - - 240^4 = 9068580571 - - - - - 241^17 = 13004045550202214485...57 <39> 241^31 = 28994597087024312133...31 <72> - - - - - 242^1 = 157 242^5 = 967465937611 242^85 = 50522793369794378938...91 <203> - - - - - 244^2 = 57737 - - - - - 245^1 = 97 245^33 = 31890566016261943265...17 <79> - - - - - 246^8 = 27746447575503721063 - - - - - 251^7 = 251059142817757 251^13 = 62779574892815645663...13 <29> 251^17 = 24918004303880136730...17 <39> 251^89 = 14894337931244879523...89 <212> 251^227 = 21281134058452601434...77 <543> 251^461 = 71063336477419750905...61 <1104> 251^3467 = 18657453422079062098...17 <8318> - - - - - 252^6 = 640129596096943 - - - - - 257^23 = 10476636392064121595...07 <54> 257^59 = 59953468189026206880...07 <140> 257^487 = 16950835022189584123...07 <1172> 257^967 = 99332255295418926667...07 <2328> 257^5657 = 38321166433624043672...01 <13631> - - - - - 260^2 = 108263 260^1514 = 31784943430378659465...83 <3657> - - - - - 263^5 = 4802611441 263^19 = 36379600869079813407...73 <44> - - - - - 265^1 = 59 265^77 = 10659671319277034195...39 <187> - - - - - 271^41 = 20911113616804317302...41 <98> 271^79 = 59321239228368075185...49 <190> 271^97 = 36868668260035434368...17 <234> 271^313 = 12246846641849019702...73 <760> 271^709 = 34983516798952751088...29 <1723> 271^829 = 31635767178958626480...49 <2015> 271^1213 = 57595170265338907426...73 <2949> - - - - - 275^1 = 97 275^13 = 19291068693956732740...69 <32> - - - - - 276^2 = 146683 276^134 = 25792424851044231987...83 <328> 276^202 = 24734876955204945717...83 <494> 276^5726 = 94423280004396105355...83 <13977> - - - - - 277^5 = 5908670381 277^19 = 92513802175220980751...87 <44> 277^109 = 61572352347698609680...61 <264> 277^1621 = 65886520281674806546...01 <3957> - - - - - 278^2 = 58943 278^3374 = 16345205189532167133...63 <8247> - - - - - 279^1 = 137 279^889 = 76295583661080535456...01 <2174> 279^1891 = 23873224377393411616...57 <4625> 279^5983 = 61703652913192164451...13 <14632> - - - - - 280^2 = 146009 280^4 = 11771915971 - - - - - 282^2 = 125863 282^94 = 43489253763584830888...03 <231> - - - - - 283^29 = 44828842143353188592...61 <69> 283^31 = 35902971384190135211...13 <74> 283^719 = 24221332618600008993...53 <1761> - - - - - 286^62 = 27265966450312501826...37 <153> 286^68 = 14921658443262036005...83 <168> - - - - - 288^2 = 164743 - - - - - 291^1 = 101 291^3 = 12245029 291^859 = 15718514671832158928...89 <2117> 291^2431 = 27301731460999158006...09 <5990> - - - - - 293^3 = 86143 293^31 = 10174507108145889152...43 <75> 293^6301 = 18513690525485261510...01 <15542> 293^7583 = 61913063505149070037...43 <18704> - - - - - 299^1 = 37 299^3 = 3542701 299^103 = 13079661808692495005...01 <255> 299^2811 = 17146461057513789284...01 <6959> - - - - - 300^2 = 224743[/code]I will probably increase the exponent range and run again, and probably the base range as well. Any direction you would like me to explore in particular? |
[QUOTE=garambois;637049]
[CODE]For base 6 = 2 * 3 for all even exponents, seems divisible by 5 For base 10 = 2 * 5 for all even exponents seems divisible by 3 For base 22 = 2 * 11 for all even exponents seems divisible by 3 For base 26 = 2 * 13 for all even exponents seems divisible by 5 For base 34 = 2 * 17 for all even exponents seems divisible by 3 For base 46 = 2 * 23 for all even exponents seems divisible by 3 For base 58 = 2 * 29 for all even exponents seems divisible by 3 For base 82 = 2 * 41 for all even exponents seems divisible by 3 For base 86 = 2 * 43 for all even exponents seems divisible by 5 For base 94 = 2 * 47 for all even exponents seems divisible by 3 For base 106 = 2 * 53 for all even exponents seems divisible by 3 For base 118 = 2 * 59 for all even exponents seems divisible by 3 For base 142 = 2 * 71 for all even exponents seems divisible by 3 For base 146 = 2 * 73 for all even exponents seems divisible by 5 For base 166 = 2 * 83 for all even exponents seems divisible by 3 For base 178 = 2 * 89 for all even exponents seems divisible by 3[/CODE] I see the results mentioned in the previous post for b = 6, b = 26. For b = 46, it seems that all s(46^e) with even e are also divisible by 3 in addition to 5. But I don't show the 5, the 3 is enough to prevent s(n^e) from being prime. I've also tried running my program for bases of the form b = 3 * q with odd exponents, here's the result : [CODE]For base 15 = 3 * 5 for all odd exponents seems divisible by 3 For base 33 = 3 * 11 for all odd exponents seems divisible by 3 For base 51 = 3 * 17 for all odd exponents seems divisible by 3 For base 69 = 3 * 23 for all odd exponents seems divisible by 3 For base 87 = 3 * 29 for all odd exponents seems divisible by 3 For base 123 = 3 * 41 for all odd exponents seems divisible by 3 For base 141 = 3 * 47 for all odd exponents seems divisible by 3 For base 159 = 3 * 53 for all odd exponents seems divisible by 3 For base 177 = 3 * 59 for all odd exponents seems divisible by 3 For base 213 = 3 * 71 for all odd exponents seems divisible by 3 For base 249 = 3 * 83 for all odd exponents seems divisible by 3 For base 267 = 3 * 89 for all odd exponents seems divisible by 3[/CODE] Here, the prime numbers q are prime numbers of the form 5 modulo 6. For bases of the form b = 5 * q with odd exponents : [CODE]For base 95 = 5 * 19 for all odd exponents seems divisible by 5 For base 145 = 5 * 29 for all odd exponents seems divisible by 5 For base 295 = 5 * 59 for all odd exponents seems divisible by 5 For base 395 = 5 * 79 for all odd exponents seems divisible by 5 For base 445 = 5 * 89 for all odd exponents seems divisible by 5[/CODE] Here, the prime numbers q are prime numbers of the form 9 modulo 10. For bases of the form b = 7 * q with odd exponents : [CODE]For base 91 = 7 * 13 for all odd exponents seems divisible by 7 For base 287 = 7 * 41 for all odd exponents seems divisible by 7 For base 581 = 7 * 83 for all odd exponents seems divisible by 7 For base 679 = 7 * 97 for all odd exponents seems divisible by 7[/CODE] Here, the prime numbers q are prime numbers of the form 13 modulo 14. [/QUOTE] For some base it's more complicated. For example, base 14 seem to be (at least empirically) covered by 2,3,5,7, and 13. Specifically, the odd exponents is divisible by 2, the exponent of form 12k,12k+4, 12k+8 (i.e., divisible by 4) is divisible by 5 the exponent of form 12k+2 is divisible by 7 the exponent of form 12k+6 is divisible by 3 the exponent of form 12k+10 is divisible by 13 I haven't try to prove it yet, but similar method to what Brownfox did is likely to work, although it might be mssier. [QUOTE] But I guess I could have deduced that from warachwe's post 3 years ago ! [/QUOTE] Looking at that post, I realized I had made some mistake in the later part. The part where I proved 79 divide s(s(n)) is depended on 157 divide s(n) exactly odd times, which may be false. |
[QUOTE=EdH;637121]I know it's a totally different format (it can be changed, if desired), but here is a listing of all the index 1 terminations I found up through exponent 10000 for everything up through base 300. This includes bases not in the tables, but excludes any squares, etc. that I noticed, like 225 and 289. I thought you may like to compare:[code]2^2 = 3
2^3 = 7 2^5 = 31 ... ... ... 299^103 = 13079661808692495005...01 <255> 299^2811 = 17146461057513789284...01 <6959> - - - - - 300^2 = 224743[/code]I will probably increase the exponent range and run again, and probably the base range as well. Any direction you would like me to explore in particular?[/QUOTE] Edwin, I'm blown away ! I don't know how you managed to achieve all this in such a short space of time. All I have to do now is shut down the 6 cores of my computer that are doing this work. It would have taken me a few more days ! Now, Thanks to you, I'll be able to finish my analysis document several days ahead of time ! To answer your question, I stopped at 200 (Limit of the exhaustive project work at this time) with the bases, no need to go beyond that. On the other hand, the further you can go with exponents, the better. I don't know how far you can go, so it's up to you ! Just one important question : What primality test do you use ? Thanks a lot Edwin. |
I have a lot more threads on a lot more machines.:smile:
I'm using YAFU's isprime() and, in fact, I'm using YAFU to do all the work, specifically:[code]#!/bin/bash base=$1 beginexp=1 endexp=10000 stepexp=1 while [ ${beginexp} -le ${endexp} ] do temp=$(./yafu "isprime(sigma(${base}^${beginexp},1)-${base}^${beginexp})" -silent -logfile /dev/null) printf "${base}^${beginexp} \r" if [ $temp -eq 1 ] then tempp=$(./yafu "sigma(${base}^${beginexp},1)-${base}^${beginexp}" -silent -logfile /dev/null) if [ ${#tempp} -gt 25 ] then echo "${base}^${beginexp} = ${tempp:0:20}...${tempp:${#tempp}-2} <${#tempp}>" echo "${base}^${beginexp} = ${tempp:0:20}...${tempp:${#tempp}-2} <${#tempp}>" >>ind1primes${base} else echo "${base}^${beginexp} = $tempp" echo "${base}^${beginexp} = $tempp" >>ind1primes${base} fi fi let beginexp=${beginexp}+${stepexp} done let beginexp=${beginexp}-${stepexp} echo "Completed through ${base}^${beginexp}."[/code]Then I run a separate script that feeds as many threads as I want on several machines. Unfortunately, this method is very inefficient and I'm trying to write a C++ program with GMP that can do better. The inefficiency is YAFU creating the original term and then factoring it to calculate the aliquot sum. If I can do the sum directly from the input, I should be able to improve for much larger exponents. As to exhaustive tables, since I'm not using the tables (or factordb) for anything, I'm not limited to where we've been for index 1 primality testing. I'm free to work anywhere of interest. Output example for the above script (YAFU must exist in the calling directory):[code]$ bash ind1pSearch.sh 2 2^2 = 3 2^3 = 7 2^5 = 31 2^7 = 127 . . . 2^4253 = 19079700752443907380...91 <1281> 2^4423 = 28554254222827961390...07 <1332> 2^9689 = 47822027880546120295...11 <2917> 2^9941 = 34608828249085121524...51 <2993> Completed through 2^10000.[/code]I'm currently running bases from 301-460 (except squares) through exponent 10000, but I will move to higher exponents on lower bases when this batch is completed. If I can get my C++ program working, it would be nice to see if I actually do improve the efficiency. |
[QUOTE=warachwe;637125]For some base it's more complicated. For example, base 14 seem to be (at least empirically) covered by 2,3,5,7, and 13. Specifically,
the odd exponents is divisible by 2, the exponent of form 12k,12k+4, 12k+8 (i.e., divisible by 4) is divisible by 5 the exponent of form 12k+2 is divisible by 7 the exponent of form 12k+6 is divisible by 3 the exponent of form 12k+10 is divisible by 13 I haven't try to prove it yet, but similar method to what Brownfox did is likely to work, although it might be mssier. [/QUOTE] Unfortunately, my little program doesn't allow me to include bases of the form B = 2 * q (q prime) in the little rule I've established empirically in my post #2402 (For all bases of the form b = p * q ([B]with 2 < p < q[/B]), the s((p * q)^e) all seem divisible by p if q is a prime such that q == p - 1 modulo 2p.). That's why I wrote "with 2 < p < q" and not "with 2 <= p < q". The case of B = 2 * q follows different rules which I have a problem with, as you say for example for B = 2 * 7. But other cases are even more troublesome, such as bases of the form B = 2^i * q. For example, for B = 12 = 2^2 * 3, we have exponent 4414 which gives a sequence that ends with a prime at index 1. This kind of case seems terrible to me, because it shows us that for a base that is not prime, we can have a first very large exponent that gives us a sequence that falls on a prime at index 1. This must be unprovable ! I don't know how to tackle these problems empirically using a computer ! To give you an idea of the complexity, here's what my little program tells me for bases of the form B = 2^2 * q under the same conditions as the other tests in my post #2402 : [CODE]For base 20 = 2^2 * 5 for all even exponents seems divisible by 3 For base 44 = 2^2 * 11 for all even exponents seems divisible by 3 For base 68 = 2^2 * 17 for all even exponents seems divisible by 3 For base 76 = 2^2 * 19 for all even exponents seems divisible by 5 For base 92 = 2^2 * 23 for all even exponents seems divisible by 3 For base 116 = 2^2 * 29 for all even exponents seems divisible by 3 For base 164 = 2^2 * 41 for all even exponents seems divisible by 3 For base 188 = 2^2 * 47 for all even exponents seems divisible by 3 For base 212 = 2^2 * 53 for all even exponents seems divisible by 3 For base 236 = 2^2 * 59 for all even exponents seems divisible by 3 For base 284 = 2^2 * 71 for all even exponents seems divisible by 3 For base 316 = 2^2 * 79 for all even exponents seems divisible by 5 For base 332 = 2^2 * 83 for all even exponents seems divisible by 3 For base 356 = 2^2 * 89 for all even exponents seems divisible by 3[/CODE] If anyone manages to come up with a rule when they see this, I'd love to hear about it ! That said, looking at this data, I can see that it was a good idea to try and find an exponent that gives a prime for B = 12 = 2^2 * 3. But we can also try our luck 2^2 * 31 = 124. For 124, Edwin hasn't got anything yet, even though he's tested exponents up to 10000 ! [QUOTE=warachwe;637125] Looking at that post, I realized I had made some mistake in the later part. The part where I proved 79 divide s(s(n)) is depended on 157 divide s(n) exactly odd times, which may be false.[/QUOTE] Okay, got it. Thanks for pointing that out. Thank you very much warachwe for your interest in these problems. |
[QUOTE=EdH;637129]I'm currently running bases from 301-460 (except squares) through exponent 10000, but I will move to higher exponents on lower bases when this batch is completed. If I can get my C++ program working, it would be nice to see if I actually do improve the efficiency.[/QUOTE]
Thanks a lot Edwin for all these explanations. Your method is much more efficient than mine. I'm going to have a look at your data tomorrow anyway, to see if we have the same thing with the work that's been done so far on my machine. I don't think you need to do the calculations for bases that are prime numbers p<41, because the OEIS knows them, (especially base 2 !!!). For bases that are prime numbers 41<p<89, I don't know, because I don't have enough to interrogate the OEIS. And for bases that are primes >=89, OEIS doesn't know them. [B]If you find more than 4 or 5 exponents for these bases that are primes p>41, it's very likely that the OEIS administrators are interested. [/B] For bases that are not prime numbers, the field is totally virgin, as nobody seems to have been interested in this before us ! Otherwise, it would be fun if your program could find an exponent for base 124 (see previous post). There must be one, but it must be very large ! |
Thanks to Wolfram Alpha's formula calculator, I have a proof by induction of the conjecture (which is number 64 on [url]http://www.aliquotes.com/conjectures_mersenneforum.html[/url]) that 5 divides the first index of 6^(2i):
[code]s(2^2*3^2)-2^2*3^2 = 1+2+2^2+3+3^2+2*3+2*3^2+2^2*3 = 1+2+4+3+9+6+18+12 = 55 Assume s(2^n*3^n) = 1 mod 10, so that s(2^n*3^n) - 2^n*3^n = 5 mod 10. s(2^(n+2)*3^(n+2))-2^(n+2)*3^(n+2) = (s(2^n*3^n)) + 3*2^(n+1)*(3^(n+3)-1)/2+4*3^(n+1)*(2^(n+1)-1) - 2^(n+2)*3^(n+2). 3*2^(n+1)*(3^(n+3)-1)/2+4*3^(n+1)*(2^(n+1)-1) = 3*2^n*3^(n+3) - 3*2^n + 4*3^(n+1)*2^(n+1) - 4*3^(n+1) = 6^n*3^4 - 3*2^n + 6^(n+1)*4 - 4*3^(n+1). Taking modulo 10, this resolves to either 6 - 2 + 4 - 8 = 0 or 6 - 8 + 4 - 2 = 0. Therefore, s(2^(n+2)*3^(n+2))-2^(n+2)*3^(n+2) modulo 10 = 1 + 0 - 6 = 5. [/code] |
Thank you very much Happy for your collaboration.
At the moment, I'm having trouble understanding your demonstration. I'm going to have to spend a lot more time on it, because it's complicated for me and I hope I'll be able to understand it. [B]If someone else could also try to validate it, that would be great, so I could add it to my site.[/B] It's not that I don't trust you Happy, but it's better to have it verified. :smile: |
[QUOTE=EdH;637121]I know it's a totally different format (it can be changed, if desired), but here is a listing of all the index 1 terminations I found up through exponent 10000 for everything up through base 300. This includes bases not in the tables, but excludes any squares, etc. that I noticed, like 225 and 289. I thought you may like to compare:[code]...
... [COLOR="Red"]23^5 = 292561 23^3181 = 20597935370569888091...01 <4331> ????????????? 43^5 = 3500201 43^13 = 40911050578149780601 43^6277 = 45524625586914719910...01 <10252>[/COLOR] ... ... [/code]I will probably increase the exponent range and run again, and probably the base range as well. Any direction you would like me to explore in particular?[/QUOTE] Edwin, the results of our programs match perfectly, for all the calculations my computer has made. This is very good news. On the other hand, there seem to be several bases missing for which my program found exponents < 10000, bases 28 to 41 included, where I put the "?" . Here's what my own program shows that you seem to be missing, the bases in blue (I don't know if you're missing any more bases further on ?) : [CODE]****************************** BASE 23****************************** 5 3181 ****************************** BASE 24****************************** ****************************** BASE 26****************************** [COLOR="Blue"]****************************** BASE 28****************************** 2 ****************************** BASE 29****************************** 5 151 3719 ****************************** BASE 30****************************** 4 14 8972 ****************************** BASE 31****************************** 7 17 31 5581 9973 ****************************** BASE 33****************************** ****************************** BASE 34****************************** ****************************** BASE 35****************************** 1 ****************************** BASE 37****************************** 13 71 181 251 463 521 7321 ****************************** BASE 38****************************** 2 5726 ****************************** BASE 39****************************** 1 591 ****************************** BASE 40****************************** ****************************** BASE 41****************************** 3 83 269 409 1759 ****************************** BASE 42****************************** 34 894 4614[/COLOR] ****************************** BASE 43****************************** 5 13 6277[/CODE] So before publishing the data document for all to see, I'll wait until you've checked that you're displaying all the bases. Because I'm adding in red all the exponents <10000 (or <20000 for example if you rerun your program to go further) found by your program that the project data didn't allow to be displayed. |
I was short on time, so I just pasted my notes. Here is the nice proof.
[quote] We prove by induction that 5 divides [$]\sigma(6^n) - 6^n[/$] for all even [$]n \geq 2[/$], by showing that [$]\sigma(6^n) - 6^n \equiv 5 \pmod{10}[/$] for all such [$]n[/$]. Base step: [$]n=2[/$]: [$]\sigma(6^n) - 6^n = \sigma(6^2) - 6^2 = 1 + 2 + 2^2 + 3 + 3^2 + 2*3 + 2*3^2 + 2^2*3 = 55 \equiv 5 \pmod{10}[/$] Inductive step: Assume [$]\sigma(6^n) - 6^n \equiv 5 \pmod{10}[/$] for an even [$]n[/$]. Since [$]6^n \equiv 6 \pmod{10}[/$] for all [$]n \geq 1[/$], this implies that [$]\sigma(6^n) \equiv 1 \pmod{10}[/$] for the even [$]n[/$] under this assumption. We now show that [$]\sigma(6^{n+2}) - 6^{n+2} \equiv 5 \pmod{10}[/$]. We have [$]\sigma(6^{n+2}) = \sigma(6^n) + (2^{n+1}+2^{n+2})(\displaystyle\sum_{i=0}^{n+2} 3^i) + (3^{n+1}+3^{n+2})(\displaystyle\sum_{i=0}^{n} 2^i) = \sigma(6^n) + (3*2^{n+1})*(\frac{1}{2}(3^{n+3}-1)) + (4*3^{n+1})*(2^{n+1}-1)[/$] We prove that [$]x = (3*2^{n+1})*(\frac{1}{2}(3^{n+3}-1)) + (4*3^{n+1})*(2^{n+1}-1) \equiv 0 \pmod{10}[/$]. [$](3*2^{n+1})*(\frac{1}{2}(3^{n+3}-1)) + (4*3^{n+1})*(2^{n+1}-1) = 3*2^n*(3^{n+3}-1) + (4*3^{n+1})*(2^{n+1}-1) = ((3*2^n)*(3^{n+3}) - (3*2^n)*1) + ((4*3^{n+1})*(2^{n+1}) - (4*3^{n+1})*1) = 3^{n+4}*2^n - 3*2^n + 4*3^{n+1}*2^{n+1} - 4*3^{n+1} = 6^{n}*3^4 - 3*2^n + 4*6^{n+1} - 4*3^{n+1}[/$]. There are two cases: either [$]n = 4m[/$] for some integer [$]m[/$], or [$]n = 4m+2[/$]. If [$]n = 4m[/$], then in [$]\mathbb{Z}_{10}[/$], [$]6^{n}*3^4 - 3*2^n + 4*6^{n+1} - 4*3^{n+1} = 6*1 - 3*6 + 4*6 - 4*3 = 6 - 8 + 4 - 2 = 0 - 0 = 0[/$]. If [$]n = 4m + 2[/$], then in [$]\mathbb{Z}_{10}[/$], [$]6^{n}*3^4 - 3*2^n + 4*6^{n+1} - 4*3^{n+1} = 6*1 - 3*4 + 4*6 - 4*7 = 6 - 2 + 4 - 8 = 0 - 0 = 0[/$]. In both cases, [$]x \equiv 0 \pmod{10}[/$], and so [$]\sigma(6^{n+2}) - 6^{n+2} = \sigma(6^n) + x - 6^{n+2} \equiv 1 \text{ (inductive assumption) } + 0 - 6 \equiv 5 \pmod{10}[/$]. QED [/quote] |
[QUOTE=Happy5214;637152]Thanks to Wolfram Alpha's formula calculator, I have a proof by induction of the conjecture (which is number 64 on [url]http://www.aliquotes.com/conjectures_mersenneforum.html[/url]) that 5 divides the first index of 6^(2i):[/QUOTE]
Here is a much simpler proof. [$$]\sigma(6^{2n}) = \frac{2^{2n+1}-1}{2-1}\frac{3^{2n+1}-1}{3-1} = \frac{(2^{2n+1}-1)(3^{2n+1}-1)}{2}[/$$] [$$]\frac{(2^{2n+1}-1)(3^{2n+1}-1)}{2} \equiv \frac{(2^{2n+1}-1)((-2)^{2n+1}-1)}{2} \equiv \frac{(2^{2n+1}-1)(-2^{2n+1}-1)}{2} \equiv \frac{2}{2} \equiv 1 \pmod 5[/$$] [$$]\sigma(6^{2n})-6^{2n} \equiv 1-1 \equiv 0 \pmod 5[/$$] |
[QUOTE=charybdis;637162]Here is a much simpler proof.
[$$]\sigma(6^{2n}) = \frac{2^{2n+1}-1}{2-1}\frac{3^{2n+1}-1}{3-1} = \frac{(2^{2n+1}-1)(3^{2n+1}-1)}{2}[/$$] [$$]\frac{(2^{2n+1}-1)(3^{2n+1}-1)}{2} \equiv \frac{(2^{2n+1}-1)((-2)^{2n+1}-1)}{2} \equiv \frac{(2^{2n+1}-1)(-2^{2n+1}-1)}{2} \equiv \frac{2}{2} \equiv 1 \pmod 5[/$$] [$$]\sigma(6^{2n})-6^{2n} \equiv 1-1 \equiv 0 \pmod 5[/$$][/QUOTE] Nice! I'm a little frustrated now that I spent almost two hours on this. Can you whip up a proof on similar lines that the aliquot sums of [$]6^{2n+1}[/$] are all divisible by 6 (which merges conjecture 63 regarding divisibility by 3 and the previously mentioned even parity)? I could then add them to the conjecture page and post a modified copy. |
[QUOTE=Happy5214;637163]Can you whip up a proof on similar lines that the aliquot sums of [$]6^{2n+1}[/$] are all divisible by 6 (which merges conjecture 63 regarding divisibility by 3 and the previously mentioned even parity)?[/QUOTE]
Sure. [$$]\sigma(6^{2n+1}) = \frac{(2^{2n+2}-1)(3^{2n+2}-1)}{2} = \frac{(4^{n+1}-1)(9^{n+1}-1)}{2}[/$$] Now [$]4^{n+1}-1 \equiv 0 \pmod 3[/$] and [$]9^{n+1}-1 \equiv 0 \pmod 8[/$], so [$](4^{n+1}-1)(9^{n+1}-1)[/$] is divisible by 24 and thus [$]\sigma(6^{2n+1})[/$] is divisible by 12. Hence 6 divides [$]\sigma(6^{2n+1})-6^{2n+1}[/$] (and in fact 12 does, as long as n > 0) |
[QUOTE=garambois;637159]Edwin, the results of our programs match perfectly, for all the calculations my computer has made.
This is very good news. On the other hand, there seem to be several bases missing for which my program found exponents < 10000, bases 28 to 41 included, where I put the "?" . Here's what my own program shows that you seem to be missing, the bases in blue (I don't know if you're missing any more bases further on ?) : So before publishing the data document for all to see, I'll wait until you've checked that you're displaying all the bases. Because I'm adding in red all the exponents <10000 (or <20000 for example if you rerun your program to go further) found by your program that the project data didn't allow to be displayed.[/QUOTE]I will check on these specific bases through 10000. Since I had several machines running these, it it possible I missed the files from one of them. I have run base 124 through exponent 20000 with two distinctly separate programs and neither found a termination at index 1. I have a working C++ program, but trying it on the listed primes from 43 through 487 did not even finish through 10000 overnight for most of them, surprisingly. Perhaps it is slower than the YAFU version after all, but it might be due to a setting I used for the prime testing. Here is the output so far for base 251:[code]Running base 251 through exponent 20000. . . 251^7 (p) 251^13 (prp) 251^17 (prp) 251^89 (prp) 251^227 (prp) 251^461 (prp) 251^3467 (prp) Current exponent: 9689[/code]GMP provided the p vs. prp value. |
[QUOTE=EdH;637169]I will check on these specific bases through 10000. Since I had several machines running these, it it possible I missed the files from one of them.
. . .[/QUOTE]Indeed this was the case. These bases were on a machine that goes off for the night and therefore they weren't included. They do match your listing. I'm currently running the same set with my C++ program to see how it does. Missing bases:[code]28^2 = 983 - - - - - - - 29^5 = 732541 29^151 = 23710442077144775649...51 <220> 29^3719 = 16255343636502228458...31 <5438> - - - - - - - 30^4 = 2119531 30^14 = 1315261613208745764481 30^8972 = 14833035585917138236...31 <13254> - - - - - - - 31^7 = 917087137 31^17 = 751670559138758105956097 31^31 = 56897247102410786528...81 <45> 31^5581 = 64936972100499919812...81 <8322> 31^9973 = 74653702974961747932...13 <14872> - - - - - - - 35^1 = 13 - - - - - - - 37^13 = 6765811783780036261 37^71 = 61097003103184669235...67 <110> 37^181 = 19418226179729227864...01 <283> 37^251 = 11543306839406838676...67 <393> 37^463 = 33196763330499316018...07 <725> 37^521 = 29977522535899491490...01 <816> 37^7321 = 17722271833278104792...01 <11480> - - - - - - - 38^2 = 1223 38^5726 = 77025002026592797969...23 <9046> - - - - - - - 39^1 = 17 39^591 = 13033551430307753172...61 <941> - - - - - - - 41^3 = 1723 41^83 = 18155161494115049258...03 <133> 41^269 = 17250332331934374410...09 <433> 41^409 = 10630101394803255136...49 <659> 41^1759 = 19263889511868538994...99 <2836> - - - - - - - 42^34 = 38762866205134159952...03 <56> 42^894 = 38265344217313381585...03 <1452> 42^4614 = 11753464626154936375...03 <7491>[/code] |
I found the following, as well:[code]175^1 = 73
175^761 = 41033304563824232068...73 <1707> 175^2545 = 15596076259239949420...73 <5709> - - - - - - - 180^4 = 2877694531 180^8716 = 24802026902981431402...31 <19658> - - - - - - - 185^1 = 43 185^109 = 37682504575483761424...83 <247> 185^2737 = 50517517859578423466...63 <6205> - - - - - - - 187^1 = 29 187^11 = 1650066523399581544396157 - - - - - - - 189^1 = 131 189^21 = 47946747409851826616...31 <48> 189^255 = 23595073121637055968...51 <581> 189^789 = 10079169670521016372...31 <1797> - - - - - - - 190^428 = 33196379339531166200...11 <976> 190^2504 = 16351827491479604983...71 <5707> - - - - - - - 198^50 = 15667054316475611248...23 <116> - - - - - - - 200^79 = 90669436471097188102...33 <182> 200^2243 = 24342856187127386669...13 <5162>[/code] |
I have found a few above exponent 10000 for bases <200. The search to 20000 is far from complete and I may well close it down because it is taking too long. But, here are a few, if of interest:[code]59^12251 (prp)
193^11171 (prp) 197^11719 (prp)[/code] |
@Happy and charybdis :
Thank you both so much, I totally understand what you did. Congrats ! I'm really short of time myself, but I'll put this demonstration on my site soon. |
Edwin,
Thanks for the missing bases ! Yes, I had realized that there were still others missing. I let my programs run to see if everything matches between our two programs. But everything seems perfect so far. Thanks for the attempt at base 124. Decidedly, that first exponent we're looking for for this base must be huge ! [QUOTE=EdH;637182]I have found a few above exponent 10000 for bases <200. The search to 20000 is far from complete and I may well close it down because it is taking too long. But, here are a few, if of interest:[code]59^12251 (prp) 193^11171 (prp) 197^11719 (prp)[/code][/QUOTE] I suspected that the work would be almost impossible to continue beyond exponent 10000. Don't worry, Edwin, if you stop at 10,000. Thanks to your calculations, I now know exactly where the OEIS stops for exponents of bases that are prime numbers and end at the index 1. The prime number 51 is the last one whose exponents have been entered into the OEIS, because the OEIS knows the sequence 11, 31, 41, 1571, [COLOR="Blue"]25771, 181981[/COLOR] ([URL="https://oeis.org/A173767"]see here[/URL]). Of course, we didn't have the last two in blue ;-) ! On the other hand, for the next prime number, 59, OEIS doesn't know the sequence 3, 13, 479. I'm thinking of contacting Michel Marcus, a Frenchman who's very active on OEIS, to ask him whether we should propose exponent sequences for bases that are prime numbers and maybe even for non-prime bases. And also how many exponents are required as a minimum for this to be accepted, but I believe this minimum is set at 4. We'll see what he says ! |
Thanks! I'm still playing with my program to see if it can be improved. In my limited testing, it does outperform the script using YAFU for smaller exponents, but I'm wondering about larger. I find the top OEIS listing for 53 interesting, as well.
As for other primes, I might try making a variation of my program that specializes in primes and see if it can knock the time to complete down even further. As for OEIS candidates, perhaps base 271 has potential:[code]Running base 271 through exponent 20000. . . 271^41 (prp) 271^79 (prp) 271^97 (prp) 271^313 (prp) 271^709 (prp) 271^829 (prp) 271^1213 (prp) Current exponent: 11423[/code] |
Two interesting (to me) items from the OEIS page referenced:
1 - I verified the 25771 exponent 2 - The last two elements were added yesterday. A third item is that this reference gives a whole new direction for searching. . . |
1 Attachment(s)
Attached is an updated copy of the conjecture list. I also took the liberty of proving conjecture 65 and converting most of the exponents to [c]<sup>[/c] tags.
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[QUOTE=Happy5214;637209]Attached is an updated copy of the conjecture list. I also took the liberty of proving conjecture 65 and converting most of the exponents to [c]<sup>[/c] tags.[/QUOTE]
Many thanks Happy for this work, it will save me time. Thanks for the corrections, I know how thorough you are ! New version online ! |
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