Sierp base 3 reservations/statuses/primes - Page 16

gd_barnes · 2009-03-24, 15:40

That is VERY strange! Can any technical person help us with this apparent PFGW problem here?

If I don't get a response within a couple of days, I'll create a separate thread for this and perhaps PM a few more knowledgeable folks.

Gary

gd_barnes · 2009-04-17, 06:50

I have not had time to do the follow ups with the particular issue involving PFGW. It is very strange for such a small test. The problem test was:

200212800*3^4+1, which factors to 19*31*41*61*101*109. Perhaps the large # of factors (i.e. smoothness) contributes to PFGW's problem. Therefore, I needed to check the k for myself using Alpertron's site and found that:

200212800*3^13+1 is prime.

For k=200M-300M and n<=25K, KEP went ahead and sent me his primes and k's remaining for n>2500 quite a while ago. I'll add k's remaining to the pages and show the range complete here shortly.

Gary

gd_barnes · 2009-04-17, 06:58

For historical balancing reference as sent to me by KEP:

k=200M-300M has 715 k's remaining at n=25K.

Subtract 246 k's that are divisible by 3.

Add back 11 k's that are divisible by 3 where k+1 is prime.

Total 480 k's remaining at n=25K.

Gary

KEP · 2009-04-17, 10:35

@ Gary regarding my base 3 reservation:

Glad that you decided to accept the range as complete. And I guess you actually may be on to something, regarding the amount of factors, because maybe it reaches a limitation in an allowed length of factors found (or how to describe it). Don't know

Off topic here, I can tell you, that I followed you advice on looking at the patterns on the riesel side, and I've actually been able to come up with a k == "remainder" mod "divisor" (divisor) answer in stead of just writing the apparent general factor for each k remaining. So in the next batches of completion I'll hand over to you, you'll get the modulos (or whatever you call it) in stead of just the factor

I can btw not tell when I will hand over the next batch of completed bases to n=25K or proven, but I can tell that I'll hand over 84 bases which you can add in your own pace :) Also it is actually quite educational for me to do the investigation myself and having to do a lot of the thinking myself, so I actually thinks that my skills will improve drasticly as I move down the list of bases

Take care everyone.

Kenneth!

gd_barnes · 2009-04-17, 11:39

You haven't been specific enough to show that you understand which k's have trivial factors.

The pattern of the trivial factors for each base is as follows:

1. Find the prime factors of the (base minus 1). Let's say they are A and B.

2. The k's with trivial factors for your base are:
(a) For Riesel, k==(1 mod A) and k==(1 mod B).
(b) For Sierp, k==(A-1 mod A) and k==(B-1 mod B).

Example for base 46:

1. The prime factors of 45 are 3*3*5.

2. The k's with trivial factors for base 46 are:
(a) For Riesel, k==(1 mod 3) and k==(1 mod 5).
(b) for Sierp, k==(2 mod 3) and k==(4 mod 5).

For bases where the base-1 is prime, it's simply k==(1 mod b-1) for Riesel and k==(b-2 mod b-1) for Sierp. Example: base 48. Riesel has k's with trivial factors for k==(1 mod 47) and Sierp has k's with trivial factors for k==(46 mod 47).

That is why base 2 is the only base with no k's that have trivial factors. If you subtract 1 from the base you get 1 but 1 is not considered prime and therefore cannot be a prime factor. Hence there are no base 2 k's with trivial factors.

A more complex example is base 211. 210 factors to 2*3*5*7, therefore many k's have trivial factors for base 211. For Riesel, it's k==1mod2, 1mod3, 1mod5, and 1mod7. For Sierp, it's k==1mod2, 2mod3, 4mod5, and 6mod7.

Does that make sense?

It's quite simple when you see the pattern. You don't have to use srsieve or Alperton's site to eliminate k's or come up with factors.

k's with algebraic factors on the Riesel side have a more complex, yet easily discernable, pattern once you are able to see it. Before I was able to pick it up, the project had to expand beyond base 32. Once it got near base 50-60, I saw 1-2 recurring patterns and then was able to "extrapolate" several more, which subsequent testing bore out.

Gary

KEP · 2009-04-17, 12:45

Actually it does make sence. Thanks for the explanaition, as soon as I'm done replying here, I'll copy this information to a notepad document and then I'll use the information, to doublecheck weather or not I infact has come up with all k's with trivial factors. Also you're right, I was not quite clear enough on what I understood, so in short term, I understood at the time I posted that the trivial factors appears in a pattern i.e. k = = 1 mod 31 (31) will appear for bases 32, 63, 94, 125, 156, 187, 218, 249 etc. etc. etc.

But now I think that I'll prepare a spreadsheet that can help me find the prime-factors of b-1

Thanks for the very educational and informative knowledge.

Regards

Kenneth

MyDogBuster · 2009-08-25, 01:45

Sierp Base 3

k=200M-300M (480k's)

Testing from n=25K-50K

MyDogBuster · 2009-10-27, 19:04

Sierp base 3 k=200M-300M-n=25K-80K complete

322 primes found and proven - see attached list

I will continue to n=100M

gd_barnes · 2009-10-27, 23:52

Quote:

Originally Posted by MyDogBuster

Sierp base 3 k=200M-300M-n=25K-80K complete

322 primes found and proven - see attached list

Nice! We "only" have 449 k's remaining for Sierp base 3 for k<300M plus 2 large squared k's > 1G.

For balancing purposes for k=200M-300M at n=80K, we have 480 - 322 = 158 k's remaining.

Quote:

Originally Posted by MyDogBuster

I will continue to n=100M

Dang! Gonna be crunching a few years? You sure you'll live that long? If you're going to test that high, I'd suggest doing k<100M first.

MyDogBuster · 2009-10-28, 00:29

Quote:

Dang! Gonna be crunching a few years? You sure you'll live that long? If you're going to test that high, I'd suggest doing k<100M first.

I meant 100K. M's and K's, it's only a letter anyway.

MyDogBuster · 2009-12-01, 23:42

Sierp base 3 k=200M-300M-n=80K-100K complete

21 primes found and proven - see attached list

Releasing range

Quote:

266356276*3^82244+1
276198764*3^82641+1
211424498*3^82946+1
268894196*3^86144+1
214111278*3^86550+1
205194422*3^87605+1
252794000*3^89442+1
208564828*3^90039+1
225771878*3^90355+1
231659672*3^90957+1
261865738*3^91446+1
289623476*3^92036+1
261518924*3^92163+1
205915818*3^92454+1
245416376*3^92788+1
249835154*3^94858+1
251552506*3^96033+1
233864102*3^96941+1
209667754*3^97143+1
221059934*3^98109+1
269558696*3^98441+1

2009-03-24, 15:40	#166
gd_barnes May 2007 Kansas; USA 2²×5×11×47 Posts	Problem with PFGW?... That is VERY strange! Can any technical person help us with this apparent PFGW problem here? If I don't get a response within a couple of days, I'll create a separate thread for this and perhaps PM a few more knowledgeable folks. Gary

2009-04-17, 06:50	#167
gd_barnes May 2007 Kansas; USA 2²·5·11·47 Posts	I have not had time to do the follow ups with the particular issue involving PFGW. It is very strange for such a small test. The problem test was: 2002128003^4+1, which factors to 19314161101109. Perhaps the large # of factors (i.e. smoothness) contributes to PFGW's problem. Therefore, I needed to check the k for myself using Alpertron's site and found that: 2002128003^13+1 is prime. For k=200M-300M and n<=25K, KEP went ahead and sent me his primes and k's remaining for n>2500 quite a while ago. I'll add k's remaining to the pages and show the range complete here shortly. Gary Last fiddled with by gd_barnes on 2009-04-17 at 06:55*

2009-08-25, 01:45	#172
MyDogBuster May 2008 Wilmington, DE 2²·23·31 Posts	Sierp Base 3 Sierp Base 3 k=200M-300M (480k's) Testing from n=25K-50K

2009-10-27, 19:04	#173
MyDogBuster May 2008 Wilmington, DE 2²·23·31 Posts	Sierp Base 3 Sierp base 3 k=200M-300M-n=25K-80K complete 322 primes found and proven - see attached list I will continue to n=100M Last fiddled with by MyDogBuster on 2014-09-02 at 09:16

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2009-04-17, 06:58	#168
gd_barnes May 2007 Kansas; USA 10340₁₀ Posts	For historical balancing reference as sent to me by KEP: k=200M-300M has 715 k's remaining at n=25K. Subtract 246 k's that are divisible by 3. Add back 11 k's that are divisible by 3 where k+1 is prime. Total 480 k's remaining at n=25K. Gary

2009-04-17, 10:35	#169
KEP Quasi Admin Thing May 2005 2²·241 Posts	@ Gary regarding my base 3 reservation: Glad that you decided to accept the range as complete. And I guess you actually may be on to something, regarding the amount of factors, because maybe it reaches a limitation in an allowed length of factors found (or how to describe it). Don't know Off topic here, I can tell you, that I followed you advice on looking at the patterns on the riesel side, and I've actually been able to come up with a k == "remainder" mod "divisor" (divisor) answer in stead of just writing the apparent general factor for each k remaining. So in the next batches of completion I'll hand over to you, you'll get the modulos (or whatever you call it) in stead of just the factor I can btw not tell when I will hand over the next batch of completed bases to n=25K or proven, but I can tell that I'll hand over 84 bases which you can add in your own pace :) Also it is actually quite educational for me to do the investigation myself and having to do a lot of the thinking myself, so I actually thinks that my skills will improve drasticly as I move down the list of bases Take care everyone. Kenneth!

2009-04-17, 11:39	#170
gd_barnes May 2007 Kansas; USA 2²·5·11·47 Posts	You haven't been specific enough to show that you understand which k's have trivial factors. The pattern of the trivial factors for each base is as follows: 1. Find the prime factors of the (base minus 1). Let's say they are A and B. 2. The k's with trivial factors for your base are: (a) For Riesel, k==(1 mod A) and k==(1 mod B). (b) For Sierp, k==(A-1 mod A) and k==(B-1 mod B). Example for base 46: 1. The prime factors of 45 are 335. 2. The k's with trivial factors for base 46 are: (a) For Riesel, k==(1 mod 3) and k==(1 mod 5). (b) for Sierp, k==(2 mod 3) and k==(4 mod 5). For bases where the base-1 is prime, it's simply k==(1 mod b-1) for Riesel and k==(b-2 mod b-1) for Sierp. Example: base 48. Riesel has k's with trivial factors for k==(1 mod 47) and Sierp has k's with trivial factors for k==(46 mod 47). That is why base 2 is the only base with no k's that have trivial factors. If you subtract 1 from the base you get 1 but 1 is not considered prime and therefore cannot be a prime factor. Hence there are no base 2 k's with trivial factors. A more complex example is base 211. 210 factors to 235*7, therefore many k's have trivial factors for base 211. For Riesel, it's k==1mod2, 1mod3, 1mod5, and 1mod7. For Sierp, it's k==1mod2, 2mod3, 4mod5, and 6mod7. Does that make sense? It's quite simple when you see the pattern. You don't have to use srsieve or Alperton's site to eliminate k's or come up with factors. k's with algebraic factors on the Riesel side have a more complex, yet easily discernable, pattern once you are able to see it. Before I was able to pick it up, the project had to expand beyond base 32. Once it got near base 50-60, I saw 1-2 recurring patterns and then was able to "extrapolate" several more, which subsequent testing bore out. Gary

2009-04-17, 12:45	#171
KEP Quasi Admin Thing May 2005 964₁₀ Posts	Actually it does make sence. Thanks for the explanaition, as soon as I'm done replying here, I'll copy this information to a notepad document and then I'll use the information, to doublecheck weather or not I infact has come up with all k's with trivial factors. Also you're right, I was not quite clear enough on what I understood, so in short term, I understood at the time I posted that the trivial factors appears in a pattern i.e. k = = 1 mod 31 (31) will appear for bases 32, 63, 94, 125, 156, 187, 218, 249 etc. etc. etc. But now I think that I'll prepare a spreadsheet that can help me find the prime-factors of b-1 Thanks for the very educational and informative knowledge. Regards Kenneth