Sierpinski/Riesel Base 5: Post Primes Here
I have been looking for the lowest Sierpinski and Riesel numbers base 5. In the y*hoo primeform group you will see posts about it.
The lowest even Reisel number is mooted to be 346802 and the lowest even Sierpinski 159986. That is to say 346802.5^n1 is always composite for every n, and 159986.5^n+1 is similarly endowed. The Sierpinskis look the easiest to look at. We have to show there is a prime for every even k (in the power series k.5^n+1) less than k=159986 and I have checked so far up to n=18468 for all of the remaining values of k shown below. All other values of k have a prime at less than n=18468 If anyone wants to take a stab at any of the values then reserve a k by answering this thread and run it up to a certain value of n, and post either the prp or the range tested. As usual, I will not be able to take a terribly active role in all of this, I have no computer power and work takes up a lot of my time right now. Regards to all of the mersenneforum Robert Smith [code] 2822 3706 4276 4738 5048 5114 5504 6082 6436 6772 7528 8644 9248 10918 12988 14110 15274 15506 15802 18530 21380 23690 24032 25240 25570 26798 27520 27676 29356 29914 30410 30658 31286 31712 32122 32180 32518 33358 33448 33526 33860 34094 36412 37246 37292 37328 37640 37714 37718 38084 40078 41738 42004 43018 43220 44134 44312 44348 44738 45652 45748 46240 46922 48424 49804 50192 51176 51208 51460 54590 55154 57316 58642 59302 59444 59912 60124 60394 60722 62698 64258 64940 66242 67282 67612 67748 68294 68416 68492 70550 71098 71492 74632 76246 76324 76370 76724 77072 77530 77908 78002 78398 79010 81556 81674 81700 82486 83032 83936 84032 84284 86354 89806 90056 90676 92158 92162 92182 92650 92906 93254 93374 93484 95246 96806 96994 98288 99784 99926 100898 101152 101284 102196 102482 104624 105166 105464 105754 106418 106588 106688 106900 107216 107258 108074 108308 109208 109988 110242 110488 110846 111382 111424 111502 111688 111994 113156 114158 117434 118388 118450 118568 120160 123406 123748 123910 125494 126134 126200 127312 127850 128432 128552 128896 129028 130324 131416 132028 133778 133990 135376 135526 136408 137132 137422 137600 137714 138022 138380 138514 138724 139196 139394 139606 139784 140498 141532 141578 143092 144052 144932 145982 146372 146780 149570 150506 150526 152050 152588 152836 153290 154222 156430 158044 158560 158696 159106 159128 159706 [/code] 
5 less to look for
89038*5^18576+1
79010*5^18901+1 15802*5^18902+1 106588*5^18920+1 82486*5^19224+1 
And more  204 to go!
81700*5^20040+1
89806*5^20852+1 95246*5^21669+1 132028*5^21736+1 138022*5^22280+1 141532*5^22472+1 The size of these firsttime prp numbers is getting interesting and only 204 candidates to check. Regards Robert Smith 
Sieve
Actually have not started sieving yet! Just pre factoring ( f100) in pfgw. Populations of candidates with no factors under 1000000 are only 13% or so in any case.
Don't know of a quick way to sieve 200 candidates, Phil Carmody developed software to sieve 12 at a time. But it would be easy to take a few candidates and sieve in NewPGen and run up to 200000 or so. Regards Robert Smith PS: 150506*5^22667+1 popped up overnight 
A first result using winpfgw (v1.2 rc1b)
159128*5^19709+1 Lars 
Some more
Lars and other interested folk:
I have been away for a few days, and the following are all first primes  sorry for the one over 150000, I shall stop testing these. 150506*5^22667+1 33860*5^23213+1 6772*5^23214+1 104624*5^23443+1 137714*5^23863+1 96806*5^24813+1 106418*5^25077+1 59302*5^25228+1 All remaining candidates are tested to n=25228. After these and Lars's discovery there are 195 remaining candidates. I will take on the first 12 of the remainder and check to n=200000, namely 2822 3706 4276 4738 5048 5114 5504 6082 6436 7528 8644 9248 Regards Robert Smith 
I found the following two primes:
15274*5^31410+1 15506*5^39203+1 These four k have no primes for n < 50000: 10918 12988 14110 18530 I am going to stop there for now. 
I found these two primes: 30658*5^29860+1, 31286*5^59705+1 (196 to go).
These two k have no primes for n < 60000, so I'm releasing them: 30410, 31712. I'm reserving the following six k: 32122, 32180, 32518, 33358, 33448, 33526. 
Hi,
finally there is a new report from my side. I have checked my logs and found that i did forget to test k=153290 upto n=30000 and guess what i found: 153290*5^29859+1 is prime!!!!! I keep my other k reserved. Lars 
OK here are the results from my search.
All k are tested to n=50000. I will unreserve these k for now. And here comes the important part: 159706*5^35244+1 is prime 158044*5^43818+1 is prime Lars 
In the case that 21380*5^n+1 equals to 4276*5^(n+1)+1 , the following are prime:
4276*5^50626+1 21380*5^50625+1 106900*5^50624+1 Shouldn't we remove all multiples of 10 (which are multiples of 5) which have duplicate k's in the list? Like the k I mentioned above? 2822 / 14110 / 70550 18530 / 92650 4738 / 23690 / 118450 5114 / 25570 / 127850 5504 / 27520 / 137600 6082 / 30410 / 152050 6436 / 32180 6772 / 33860 And so on....The most left number is the 'base' number and the numbers following it are multiples of 5 of it. So why would we check for their primality, if we new the primality of a multiple of it? Am I right? (Just a n00b on primality) Also, I'm now reserving k = 24032 until n=100000 
You're right Geoff.
I found a prime for 21380*5^50625+1 , which is equal to 4276*5^50626+1. But this doesn't mean that k=4276 doesn't have a prime for n less than 50626, so that's one thing that has to be checked! 
[QUOTE=Templus]I found a prime for 21380*5^50625+1 , which is equal to 4276*5^50626+1. But this doesn't mean that k=4276 doesn't have a prime for n less than 50626, so that's one thing that has to be checked![/QUOTE]
Nice one! It doesn't matter for the project whether or not k=4276 could have been eliminated by a smaller n than n=50626, any prime will do. The only problems are for k such as k=123910=5*24782. 24782 has already been eliminated because 24782*5^1+1 is prime, but this doesn't rule out the possibility that 123910*5^n+1 = 24782*5^(n+1)+1 is composite for all n. This means we have to leave k=123910 in the list. 
OK the outcome of the observation by Templus is that all multiples of 5 can be eliminated except for 51460, 81700 and 123910, and Robert already found a prime for 81700. This means there are only 161 candidates left to test.

Results
4276*5^50626+1
4738*5^41656+1 5048*5^37597+1 5504*5^39475+1 are all PRP3. other checked to 2822 50057 3706 65328 5114 191771 Will now start on: 6082 6436 7528 8644 9248 Regards Robert Smith 
[QUOTE=Templus]Geoff, did you see that I reserved k = 24032 on the sixth of january?[/QUOTE]
Sorry I missed that, noted now. My new results are: 33358*5^38096+1 and 33526*5^41142+1 are prime. 
Results to 10000
Searching for the remaining candidates k less than 10000 did not reveal any new prps:
K largest n checked 6082 77402 6436 61512 7528 90216 8644 79150 9248 85471 Regards Robert Smith 
Reservations
Geoff
I will take candidates 110000120000 next Regards Robert Smith 
Seriously big prime
Now we are in business:
[url]http://primes.utm.edu/primes/page.php?id=73175[/url] Primality testing 111502*5^134008+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 Calling BrillhartLehmerSelfridge with factored part 99.99% 111502*5^134008+1 is prime! (1503.0378s+0.0176s) First prime I have found for a while. It will be the 1000 to 1100 range of largest primes ever found, tantalisingly close to 100000 digits. Interestingly this is the k value which we might have expected to give the most problem having the smallest smallest Nash weight of all the remaining candidates! Regards Robert Smith 
one down
Hi there,
My first prime for this project: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 37246*5^50452+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 Running N1 test using base 13 Calling BrillhartLehmerSelfridge with factored part 99.99% 37246*5^50452+1 is prime! (456.9443s+0.0070s) Cheers, Micha Fleuren 
One more down
Hi all,
I got one more down today, finding my second prime: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 38084*5^29705+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 Calling BrillhartLehmerSelfridge with factored part 99.98% 38084*5^29705+1 is prime! (65.7210s+0.0034s) Cheers, Micha 
The Riesel base 5 series
I have taken a slight excursion away from Sierpinski base 5 to prepare the groundwork for the Riesel base 5 study. I have checked up to around n=12250 and I am still clearing 910 candidates a day. I will stop when sieving individual candidates makes sense. Right now there are 465 candidates left, so we should still work on the Sierpinski set.
For the Sierpinski series, I have checked the following k to the following n with no primes: k n 110242 52766 110488 55772 And I have discovered: 111994 30446 is prp3 Regards Robert Smith 
Robert, could you make available a list of the primes you found for n <= 18468? Or if you email it to me ( geoff AT hisplace DOT co DOT nz ) I will make it available.
I will keep a list of primes that make the top 5000 list in [url]http://www.geocities.com/g_w_reynolds/Sierpinski5/champs.txt[/url], there is just one entry so far. My results: I found (a while ago) that 32518*5^47330+1 is prime. I am reserving these k: 10918, 12988, 31712. 
Hello all,
I agree with uncwilly that this project should deserve it's own private place... anyone know how to move it? In the meantime: One more down: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 42004*5^27992+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 Calling BrillhartLehmerSelfridge with factored part 99.98% 42004*5^27992+1 is prime! (57.0569s+0.0044s) 
Yet another one down
Hi all,
4th prime on here: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 44134*5^39614+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 7 Calling BrillhartLehmerSelfridge with factored part 99.98% 44134*5^39614+1 is prime! (150.2650s+0.0062s) Cheers, Micha 
Hi,
here my newest results: 60124*5^38286+1 is prime! 60394 tested to 50166 ( i keep this reserved) 60722 tested to 49329 ( i keep this reserved) I also keep my other k reserved. Lars 
The primes.txt file in [url]http://www.geocities.com/g_w_reynolds/Sierpinski5/[/url] contains the k,n pairs for all the primes k*5^n+1 found so far. It can be used as input to Proth.exe in file mode, or by adding 'ABC $a*5^$b+1' to the top, as input to pfgw.

Nought
This may come to nought, actually no, it will come to k+1
What am I talking about? n=0 > k*5^0+1= k+1 Therefore any k still remaining where k+1 is prime can be eliminated, unless that is not in the Sierpinski rule book. Looking at my original list, this would eliminate 7528 and maybe others....not got a list of primes to hand What does this group think of this wheeze? Regards Robert Smith 
n=0
Following on, now I am home:
From the original list n=0 eliminates (and a number of these we have found already higher primes or prp for): 7528 15802 33358 43018 51460 81700 82486 90676 102196 105166 123406 123910 143092 152836 159706 Regards Robert Smith 
[QUOTE=robert44444uk]Therefore any k still remaining where k+1 is prime can be eliminated, unless that is not in the Sierpinski rule book.[/QUOTE]
The definitions of Sierpinski number that I have seen take n to be positive, e.g. [url]http://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html[/url], so I think we should leave these k in the list. (If they turn out to be harder than the other k then we could reconsider :) 
[QUOTE=geoff]The definitions of Sierpinski number that I have seen take n to be positive, e.g. [url]http://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html[/url], so I think we should leave these k in the list. (If they turn out to be harder than the other k then we could reconsider :)[/QUOTE]
Yves Gallot's site also lists n as having to be greater than or equal to one. [url]http://www.prothsearch.net/sierp.html[/url] 
[QUOTE=pcco74]Yves Gallot's site also lists n as having to be greater than or equal to one. [url]http://www.prothsearch.net/sierp.html[/url][/QUOTE]
This is for base 2 only, 5^0 should be considered too. 
Odds and evens
I posted the following message on Yahoo primenumbers to see whether one of the maths bods can give an answer. I am reasonably confident we should allow n=0
[url]http://groups.yahoo.com/group/primenumbers/message/16018[/url] Regards Robert Smith 
The reason k must be odd for base 2 is that if k is even, you can always divide it by 2 until you get an odd k, and increase n accordingly. Ex. 10*2^2+1=5*2^3+1. This simply eliminates testing the same number multiple times, and provides for a common format for these numbers. As far as n=0, I really think this should not be included for the following two reasons. First, it is not included in the original Sierpinski numbers, which we are trying to represent in base 5. Second, including n=0 eliminates all information on the base. For example, 4*2^0+1=4*5^0+1=4*45569^0+1. This defeats the purpose by reducing the expression k*b^n+1 to the much more general form k+1, or basically k.

Update:
Primality testing 123910*5^136268+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 123910*5^136268+1 is prime! (1597.4078s+0.0268s) Yes I know that Robert believes that we don't need to test this number, but as I had sieved fairly deeply and had already done so much PRP testing, I decided to continue knowing that if I found a prime it would be near 100000 digits. It is, at 95253 digits. It will be stored as 24782*5^136269+1 in Chris Caldwell's database. 109208 done to 188000 (still reserved) 71492 done to 55000 (still reserved) 
Wow
Rogue  many congrats are in order for finding such a large prime  all power to you.
Actually though, and I am being pedantic, it is the second prime for this value of k, the first being n=0, as is shown on Geoff's list 24782 when n=1 removed that number from checking. But how nice to find a juicy big prime !! Regards Robert Smith 
Delighted
Hello all,
After a delightfull stay in Disney Resort Paris, I came home to a computer stating me a delightful find: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 37718*5^104499+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 11 Calling BrillhartLehmerSelfridge with factored part 99.99% 37718*5^104499+1 is prime! (1165.4437s+0.0267s) Number was ranked 1737th largest prime ever yesterday evening Cheers, Micha 
Congratulations rogue and michaf for the big primes.
If n=0 is allowed in the definition of base5 Sierpinski number, then k=7528 has a prime 7528*5^0+1, but we will still need to eliminate k=5*7528=37640, so a prime 7528*5^n+1 for n >=1 must be found either way. This is the only exceptional case in Roberts list above, I have added an asterisk beside the other candidates in status.txt that don't need to be tested if n=0 is allowed. My own results: 83032*5^39408+1 is prime. 33448*5^n+1 is not prime for n <= 100,000 and I am releasing it. 
One more down.
67282*5^45336+1 is prime. Lars 
And the next result:
68294*5^33723+1 is prime! I keep the rest of my ranges reserved. Lars 
six down for me now
Hya's
found myself my sixth prime: PFGW Version 20041020.Win_Stable (v1.2 RC1c) [FFT v23.8] Primality testing 46922*5^37483+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 46922*5^37483+1 is prime! (155.2515s+0.0060s) Cheers, Micha 
k=34094
34094*5^27305+1 is prime!!! [19090 digits]
Resevering k=26798 and 27676 
Hey, 51460*5^50468+1 is prime. (35281 digits)
Primality testing 51460*5^50468+1 [N1, BrillhartLehmerSelfridge] Running N1 test using base 2 Calling BrillhartLehmerSelfridge with factored part 99.99% 51460*5^50468+1 is prime! (917.7062s+0.0104s) 
Hi,
my next update. 68416*5^44578+1 is prime. All my ranges are tested to at least n=59000. I will keep them reserved. Lars 
Moving out
Group
It appears it is time to graduate from "fun stuff" but the thing is how? It seems to me there is a gentlemenly protocol here. I think we need to be invited to another forum, maybe "prime sierpinski project" but we need to be invited as we would not like people to think we are stealing their valued members and resources. Or we could start a new forum in preparation for the Riesel exercise that Geoff and I are preparing the group for. And then there is the question of moving all the messages out, rather than just starting again. If anyone has an idea please post it, or even better move the threads to a nice new home. Regards Robert Smith 
If a new forum is created I think it is possible to just move this thread there, I think that is how some other projects started. I volunteer to be a moderator if needed.
results: 86354*5^53329+1 is prime. releasing: 83936, 84284 (no primes for n <= 100,000). reserving: 90056, 90676 
Decision time
I think if Geoff can organise to set up a new forum with stickies right left and centre then this is the best solution. We should have at least two threads, Sierpinski and Riesel. Geoff you should send a private message to one of the organisers of the Mersenne site to arrange it.
Here are some results for the 110000120000 range, apologies for duplications. I give up my reservations in this range. No primes and highest tested: 110242 52766 110488 55772 110846 31969 111382 55774 111424 53896 111688 51102 113156 62153 117434 64335 118388 50885 118568 139237 One prime: 111502 134008 111994 30446 114158 45859 118450 41654 I reserve candidates in the 140000 to 150000 range. Regards Robert Smith 
New Forum
I have created the sticky reservation threads, so the forum is functional now. Please check your existing reservations and let me know if there are any mistakes.
Still needed are posts to explain to newcomers what the project is about, how to get started, etc.. If anyone can help with this then please do, otherwise I will get around to it in the next week. What do you think is the best way to record the primes found so far? I will keep the text files at [url]http://www.geocities.com/g_w_reynolds/Sierpinski5/[/url] and [url]http://www.geocities.com/g_w_reynolds/Riesel5/[/url] updated for now, but maybe a results thread would be better? Any other suggestions for the forum, let me know. 
129028*5^32462+1 is prime!

Congratulations masser, this prime made it into the top 5000:
[quote=masser (in the reservations thread)] Found the following prime: 76246*5^83568+1 [/quote] [url=http://primes.utm.edu/primes/page.php?id=74149]link[/url] 
128896*5^89436+1 is prime!

159106*5^89982+1 is prime!!

53858*5^337601 is prime! 23602 digits

98288*5^42133+1 is Prime!

77908*5^47338+1 is Prime!

78002*5^40115+1 is Prime!

Please post reservations and results etc. in the reservations thread, it makes it easier for me to keep track of them, and also easier to see what is still available if others make a reservation before I get around to updating the table.
Post a copy of the primes found to this thread if you want, as I will delete the messages in the reservations thread when the tables are updated. 
137336*5^477041 is prime!

53858*5^89840  1 is prime! 62801 digits
Woohoo!!! 
BTW this is a top 5000 prime:
[url]http://primes.utm.edu/primes/page.php?id=74786[/url] 
[QUOTE=CedricVonck]53858*5^89840  1 is prime! 62801 digits[/QUOTE]
You found the prime 53858*5^337601 earlier, so this k has already been eliminated... 
[QUOTE=geoff]You found the prime 53858*5^337601 earlier, so this k has already been eliminated...[/QUOTE]
?? Is this not a project like "primesearch"?? 
[QUOTE=CedricVonck]?? Is this not a project like "primesearch"??[/QUOTE]
Not quite, we are trying to find a prime of the form k*5^n+1 (resp. k*5^n1) to show that k is not a Sierpinski (resp. Riesel) number. One prime for each k is sufficient. I think it makes sense to record the smallest prime for each k, since this will minimise the time needed to verify the final proofs. 
203036*5^229281 is prime!
203036*5^276041 is prime! (< Redundant) 
145114*5^314591 is prime!

214958*5^202541 is prime!

136588*5^229171 is prime!

43156*5^441351 is prime!

294698*5^471101 is prime!

Another one bites the dust
Prime 9164*5^408921 Is there some other way to check these are primes. As the test has some chance of being wrong.(Am i right?) 8494 and 38348 completed upto 50k without any luck. Btw, Is there some statistics/guesses as to how many will fall till say 50k? I got 2/4. jaat 
[QUOTE=jaat]Another one bites the dust
Prime 9164*5^408921 Is there some other way to check these are primes. As the test has some chance of being wrong.(Am i right?)[/QUOTE] Yes, you can use WinPFGW to do a primality test with the tp switch (or tm for a +1 number). No, unless there is a bug in the software or if there were a hardware problem, the number is prime. 
[QUOTE=jaat]Btw, Is there some statistics/guesses as to how many will fall till say 50k?
I got 2/4.[/QUOTE] From the Sierpinski data so far I would guess that a little more than 1 in 3 candidates will be eliminated by finding a prime with an exponent between 20,000 and 50,000. The 4 candidates you worked on were of average combined weight, so I would say you did just a little better than average to eliminate 2. (I moved the last two messages from the reservations thread) 
Thanks to both Rogue and geoff for answering the queries. I checked both the numbers with the WinPFGW and they are prime. It is also nice to know that one can knock a very large fraction of them very quickly. I'll get back to them soon.
jaat 
I wondered if any of the primes we find could be one of a pair of twin primes of the form k*5^n+/1.
Since in any three consecutive integers one must be divisible by 3, for both k*5^n+1 and k*5^n1 to be prime, k must be divisible by 3. It turns out that this is possible for just one of the k we are currently testing, the Riesel candidate k=151026. Of the primes already found, the two largest twin primes of this form were 110538*5^139+/1 and 136674*5^172+/1. 
Hey all. I was wondering: Is this project still "alive"? There is not a lot going on in the forum and only geoff has found primes in the last month, so I started to wonder how many people (or GHz) are working on this problem?
I think I'll get back here when I'm finished with my 2721... 
I have a part time 2.66GHz P4 that I split between this project and PSP, it usually does 810 hours per week.
It is quiet at the moment, but that is OK, there is no rush :) 
99356*5^349941 is prime!

159622*5^215671 is prime
86762*5^222921 is prime 
319658*5^275301 is prime!

171064*5^347671 is prime!

119456*5^364241 is prime!

166186*5^382971 is prime!

45434*5^437361 is prime!

Moderation
From next week I will not have frequent internet access, so if anyone is interested in taking over as moderator then you are welcome. I don't have any scripts for maintaining the project data, I have just done it by hand, but there is not a lot of work, and after the easy primes are found it the amount of work involved will decrease.
A new moderator would need to copy of the files at [url]http://www.geocities.com/g_w_reynolds/Sierpinski5/[/url] and [url]http://www.geocities.com/g_w_reynolds/Riesel5[/url] to somewhere they have write access, and contact Xyzzy for moderator status to be able to edit the reservation threads. I am going to pack my life into a bicycle trailer and spend the next six months or so on the road or tramping through our national parks. I will log on from internet cafes when I get the chance, but I don't expect to spend a lot of time online. Keep posting your results, even if nobody picks up the moderation duties I will eventually get back to it. Good luck with the search. Geoff. 
Geoff
I am so jealous. What I would do with 6 months of the open air life. Sigh!
All the best and stay safe! Regards Robert Smith 
The following are all primes:
[CODE] 115786*5^202631 125458*5^204911 184204*5^200271 200132*5^209081 246046*5^203631 253924*5^210231 269638*5^215251 276142*5^210171 299006*5^213561 309928*5^206871 337856*5^206801[/CODE] 
152872*5^220211 is prime!

258838*5^228311 is prime!

67016*5^230701 is prime!

66412*5^233751 is prime!
132478*5^233911 is prime! 138664*5^234891 is prime! 
244388*5^238981 is prime!

86594*5^240261 is prime!
87146*5^244481 is prime! 91684*5^244591 is prime! 272794*5^248451 is prime! 
99698*5^253221 is prime!
150142*5^253251 is prime! 121438*5^254531 is prime! 
259232*5^261321 is prime!

338144*5^265541 is prime!
309046*5^265731 is prime! 208454*5^266201 is prime! 
119144*5^270361 is prime!

271868*5^273121 is prime!
66494*5^274961 is prime! PS: I have got the moderator privileges. But I'll need to figure out how to update the project status files, before I start doing active bookkeeping. 
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