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Old 2020-07-04, 16:15   #859
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Nov 2016

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These Sierpinski bases (up to 1024) cannot be proven with current knowledge and technology, since they have either GFN (for even bases) or half GFN (odd bases) remain: (half GFN is much worse, since for these (probable) primes, the divisor gcd(k+-1,b-1) is not 1 (it is 2), and when n is large (for all numbers of the form (k*b^n+-1)/gcd(k+-1,b-1) whose gcd(k+-1,b-1) is not 1) the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin test, unless a divisor of the number can be found)

Code:
b     k
2     65536
6     1296
10     100
12     12
15     225
18     18
22     22
31     1
32     4
36     1296
37     37
38     1
40     1600
42     42
50     1
52     52
55     1
58     58
60     60
62     1
63     1
66     4356
67     1
68     1
70     70
72     72
77     1
78     78
83     1
86     1
89     1
91     1
92     1
93     93
97     1
98     1
99     1
104     1
107     1
108     108
109     1
117     117
122     1
123     1
124     15376
126     15876
127     1
128     16
135     1
136     136
137     1
138     138
143     1
144     1
147     1
148     148
149     1
151     1
155     1
161     1
166     166
168     1
178     178
179     1
180     1049760000
182     1
183     1
186     1
189     1
192     192
193     193
196     196
197     1
200     1
202     1
207     1
210     1944810000
211     1
212     1
214     1
215     1
216     36
217     217
218     1
222     222
223     1
225     225
226     226
227     1
232     232
233     1
235     1
241     1
243     27
244     1
246     1
247     1
249     1
252     1
255     1
257     1
258     1
262     262
263     1
265     1
268     268
269     1
273     273
280     78400
281     1
282     282
283     1
285     1
286     1
287     1
291     1
293     1
294     1
298     1
302     1
303     1
304     1
307     1
308     1
310     310
311     1
316     316
319     1
322     1
324     1
327     1
336     336
338     1
343     49
344     1
346     346
347     1
351     1
354     1
355     1
356     1
357     357
358     358
359     1
361     361
362     1
366     366
367     1
368     1
369     1
372     372
377     1
380     1
381     381
383     1
385     385
387     1
388     388
389     1
390     1
393     393
394     1
397     397
398     1
401     1
402     1
404     1
407     1
408     408
410     1
411     1
413     1
416     1
417     1
418     418
420     176400
422     1
423     1
424     1
437     1
438     438
439     1
443     1
446     1
447     1
450     1
454     1
457     457
458     1
460     460
462     462
465     465
467     1
468     1
469     1
473     1
475     1
480     1
481     481
482     1
483     1
484     1
486     486
489     1
493     1
495     1
497     1
500     1
509     1
511     1
512     2, 4, 16
514     1
515     1
518     1
522     522
524     1
528     1
530     1
533     1
534     1
538     1
541     541
546     546
547     1
549     1
552     1
555     1
558     1
563     1
564     1
570     324900
572     1
574     1
578     1
580     1
586     586
590     1
591     1
593     1
597     1
600     129600000000
601     1
602     1
603     1
604     1
606     606
608     1
611     1
612     612
615     1
618     618
619     1
620     1
621     621
622     1
626     1
627     1
629     1
630     630
632     1
633     633
635     1
637     1
638     1
645     1
647     1
648     1
650     1
651     1
652     652
653     1
655     1
658     658
659     1
660     660
662     1
663     1
666     1
667     1
668     1
670     1
671     1
672     672
675     1
678     1
679     1
683     1
684     1
687     1
691     1
692     1
694     1
698     1
706     1
707     1
708     708
709     1
712     1
717     717
720     1
722     1
724     1
731     1
734     1
735     1
737     1
741     1
743     1
744     1
746     1
749     1
752     1
753     1
754     1
755     1
756     756
759     1
762     1
765     765
766     1
767     1
770     1
771     1
773     1
775     1
777     777
783     1
785     1
787     1
792     1
793     793
794     1
796     796
797     1
801     801
802     1
806     1
807     1
809     1
812     1
813     1
814     1
817     817
818     1
820     820
822     822
823     1
825     1
836     1
838     838
840     1
842     1
844     1
848     1
849     1
851     1
852     852
853     1
854     1
858     858
865     865
867     1
868     1
870     1
872     1
873     1
878     1
880     880
882     882
886     886
887     1
888     1
889     1
893     1
896     1
897     897
899     1
902     1
903     1
904     1
907     1
908     1
910     828100
911     1
915     1
922     1
923     1
924     1
926     1
927     1
932     1
933     933
937     1
938     1
939     1
941     1
942     1
943     1
944     1
945     1
947     1
948     1
953     1
954     1
958     1
961     1
964     1
966     870780120336
967     1
968     1
970     970
974     1
975     1
977     1
978     1
980     1
983     1
987     1
988     1
993     1
994     1
998     1
999     1
1000     10
1002     1
1003     1
1005     1005
1006     1
1008     1008
1009     1
1012     1012
1014     1
1016     1
1017     1017
1020     1020
1024     4, 16

Last fiddled with by sweety439 on 2020-07-10 at 06:28
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Old 2020-07-05, 16:11   #860
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We don't know the conjectures yet on bases 66, 120, 156, 180, 210, 240, 280, 330, 358, 420, 456, 462, 546, ... and Sierp 190, 540 and Riesel 276, 486. They are definitely k > 1M. They're just too unmanageable with today's computing power and knowledge. That said, if people are open to the idea, I might suggest searching some of those up to k=100K at some point to get future generations of prime searchers started on those huge efforts. To me, the main thing is collecting data for us and for future generations to analyze.

There are upper bounds of these conjectures: (the lower bounds of these conjectures are all 1000001, since all k<=1M are checked)

S66: 21314443 (if not this number, then must be == 4 mod 5 or == 12 mod 13)
S120: 374876369 (if not this number, then must be == 6 mod 7 or == 16 mod 17)
S156: 18406311208 (if not this number, then must be == 4 mod 5 or == 30 mod 31)
S180: 1679679 (if not this number, then must be == 178 mod 179)
S190: 3146151 (if not this number, then must be == 2 mod 3 or == 6 mod 7)
S210: 147840103 (if not this number, then must be == 10 mod 11 or == 18 mod 19)
S240: 1722187 (if not this number, then must be == 238 mod 239)
S280: 82035074042274 (if not this number, then must be == 2 mod 3 or == 30 mod 31)
S330: 16636723 (if not this number, then must be == 6 mod 7 or == 46 mod 47)
S358: 27478218 (if not this number, then must be == 2 mod 3 or == 6 mod 7 or == 16 mod 17)
S420: 2288555 (if not this number, then must be == 418 mod 419)
S456: 14836963 (if not this number, then must be == 4 mod 5 or == 6 mod 7 or == 12 mod 13)
S462: 6880642 (if not this number, then must be == 460 mod 461)
S540: 1091739 (if not this number, then must be == 6 mod 7 or == 10 mod 11)
S546: 45119296 (if not this number, then must be == 4 mod 5 or == 108 mod 109)

R66: 101954772 (if not this number, then must be == 1 mod 5 or == 1 mod 13)
R120: 166616308 (if not this number, then must be == 1 mod 7 or == 1 mod 17)
R156: 2113322677 (if not this number, then must be == 1 mod 5 or == 1 mod 31)
R180: 7674582 (if not this number, then must be == 1 mod 179)
R210: 80176412 (if not this number, then must be == 1 mod 11 or == 1 mod 19)
R240: 2952972 (if not this number, then must be == 1 mod 239)
R276: 1552307 (if not this number, then must be == 1 mod 5 or == 1 mod 11)
R280: 513613045571841 (if not this number, then must be == 1 mod 3 or == 1 mod 31)
R330: 16527822 (if not this number, then must be == 1 mod 7 or == 1 mod 47)
R358: 27606383 (if not this number, then must be == 1 mod 3 or == 1 mod 7 or == 1 mod 17)
R420: 6548233 (if not this number, then must be == 1 mod 419)
R456: 76303920 (if not this number, then must be == 1 mod 5 or == 1 mod 7 or == 1 mod 13)
R462: 2924772 (if not this number, then must be == 1 mod 461)
R486: 1525283 (if not this number, then must be == 1 mod 5 or == 1 mod 97)
Attached Files
File Type: txt Conjectured smallest Sierpinski number.txt (8.3 KB, 3 views)
File Type: txt Conjectured smallest Riesel number.txt (8.3 KB, 1 views)

Last fiddled with by sweety439 on 2020-07-10 at 16:26
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Old 2020-07-07, 08:10   #861
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Any Riesel k that is a perfect square will always be lower weight than average because the even-n have algebraic factors.
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Old 2020-07-07, 08:19   #862
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A Sierpinski form (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if k*b^n is perfect odd power A070265 or of the form 4*m^4 A101046

A Riesel form (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if k*b^n is perfect power A001597

Situations in which srsieve will make that statement:

1. For any k on any Riesel base that is a perfect square, when sieving, all n-values that are divisible by 2 can be manually removed.
2. For any k on any Sierpinski base that is of the form 4*m^4, all n-values that are divisible by 4 can be manually removed.
3. For any k on any (Riesel or Sierpinski) base that is a perfect cube, all n-values that are divisible by 3 can be manually removed.
4. (etc.) for k's that are perfect 5th powers, 7th powers, 11th powers, or any prime power where p=power, any n's divisible by p can be manually removed.

Taken to an extreme, for k=2048, which is 2^11, you could manually eliminate all n-values that are divisible by 11.

A special case is k=1:

* For Riesel base, all n-values that are not prime can be manually removed.
* For Sierpinski base, all n-values that are not power of 2 can be manually removed.

To put the above in a different way: You can only eliminate the k if manually removing all of these n-values leaves you with no n's remaining, which would have been the case had you attempted to sieve 900*67^n-1. Had you sieved it, you would have ended up with a sieve file with very few EVEN n-values remaining and zero ODD n-values remaining. Once you manually removed the n's divisible by 2, you would have had nothing left. That means that the k-value can be removed from conjecture testing because it has partial algebraic factors that combine with a numeric factor to make a full "covering set" of factors.

Analysis on both k=125 and 729 shows that they should remain because there is no factor or factors that eliminate the n's that the algebraic factors do not.

When sieving, as per the above, on k=125, you can manually remove all n-values divisible by 3. On k=729, that is one of the few that you can eliminate n-values that are divisible by 2 -or- that are divisible by 3. In effect, you're only left with n-values that are n==(1 or 5 mod 6). If you manually remove those n's, you'll stop getting that message from srsieve.

k=729 would normally be extremely low weight except for the fact that it is divisible by 3, which eliminates any possibility of a factor of 3 for all n-values. That increases the weight to something like a k that is a perfect square.

A k-value that would be extremely low-weight is one that is a perfect square and cube but is not divisible by 2 or 3. I think the lowest one of that nature would be 5^6=15625, which would only be an issue on very few bases.

Last fiddled with by sweety439 on 2020-07-07 at 08:26
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Old 2020-07-07, 08:43   #863
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Exclusions:

k's that are multiples of the base where (k+-1)/gcd(k+-1,b-1) (+ for Sierp, - for Riesel) is not prime.

I'll give some examples for S22:

k=44, 154, 220, 242, 264, 374, 440 would be excluded from testing because 45/3, 155, 221, 243/3, 265, 375/3, 441/21 are not primes.

k=22, 66, 88, 110, 132, 176, 198, 286, 308, 330, 352, 396, 418 would be INcluded in testing because 23, 67, 89, 111/3, 133/7, 177/3, 199, 287/7, 309/3, 331, 353, 397, 419 are primes.

The exclusions for multiples of the base are the same on all bases. Check for (k-1)/gcd(k+-1,b-1) being prime on the Riesel side and check for (k+1)/gcd(k+-1,b-1) being prime on the Sierp side. If it's prime, include it; if it's not prime, exclude it.

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Old 2020-07-07, 09:01   #864
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So far, there is only two Riesel k and base <= 128 where algebraic factors on even-n combine with a covering set of MORE than one factor on odd-n to eliminate a k. That is 1369*30^n-1 and (400*88^n-1)/3
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Old 2020-07-07, 09:24   #865
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(101*73^2146-1)/4 is (probable) prime!!!

R73 has only k=79 remain, reserve it to n=10K
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Old 2020-07-07, 09:32   #866
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Quote:
Originally Posted by sweety439 View Post
(101*73^2146-1)/4 is (probable) prime!!!

R73 has only k=79 remain, reserve it to n=10K
https://docs.google.com/document/d/e...ncZ4OqEDGd/pub

Newest status of Riesel problems
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Old 2020-07-07, 18:04   #867
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Checking whether a k-value makes a full covering set with algebraic factors not always very easy. The way I do it is to look for patterns in the factors of the various n-values for specific k-values. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.:
11*13
179*181
etc.

In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors.

e.g. for the case R15 k=47
n-value : factors
1 : 2^5 · 11
2 : 17 · 311
3 : 2^4 · 4957
4 : 31 · 38377
6 : 11 · 43 · 565919
8 : 199 · 1627 · 186019
10 : 17 · 61 · 13067776451
12 : 37 · 82406457849451
20 : 15061 · 236863181 · 2190492030407

Analysis:
For n=1 & 3 (and all odd n), all values are divisible by 2 so we only consider even n's.
For n=4, the two prime factors does not close.
For n=6 & 10, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors.
For n=12, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point.

R33 k=257:
n-value : factors
1 : 5 · 53
2 : 2 · 4373
3 : 397 · 727
4 : 2^2 · 2381107
5 : 5^3 · 7 · 359207
7 : 11027 · 31040117
15 : 13337 · 706661 · 51076716238627
19 : 38231 · 14932493857679888742000509
For n=15 & 19 same explanation as R15 k=47

R36 k=1555:
n-value : factors
1 : 11 · 727
2 : 31 · 37 · 251
3 : 67 · 154691
4 : 37 · 127 · 271 · 293
7 : 4943 · 3521755879
9 : 59 · 382386761790283
For n=7 & 9 same explanation as R15 k=47


The prime factors for n=12, n=15, and n=7 respectively make it clear to me that these k-values should all yield primes at some point so you are correct to include them as remaining.

The higher-math folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of n-values on them but not for all of the n-values. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime.
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Old 2020-07-07, 18:08   #868
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Quote:
Originally Posted by sweety439 View Post
Checking whether a k-value makes a full covering set with algebraic factors not always very easy. The way I do it is to look for patterns in the factors of the various n-values for specific k-values. If there are algebraic factors, it's most common for them to be in a pattern of f*(f+2), i.e.:
11*13
179*181
etc.

In other cases there may be a consistent steady increase in the differences of their factors, which is especially tricky to find but indicates the existence of algebraic factors.

e.g. for the case R15 k=47
n-value : factors
1 : 2^5 · 11
2 : 17 · 311
3 : 2^4 · 4957
4 : 31 · 38377
6 : 11 · 43 · 565919
8 : 199 · 1627 · 186019
10 : 17 · 61 · 13067776451
12 : 37 · 82406457849451
20 : 15061 · 236863181 · 2190492030407

Analysis:
For n=1 & 3 (and all odd n), all values are divisible by 2 so we only consider even n's.
For n=4, the two prime factors does not close.
For n=6 & 10, multiplying the 2 lower prime factors together does not come close to the higher prime factor so little chance of algebraic factors.
For n=12, the large lowest prime factor that bears no relation to the other prime factor means that there is unlikely to be a pattern to the occurrences of large prime factors so there must be a prime at some point.

R33 k=257:
n-value : factors
1 : 5 · 53
2 : 2 · 4373
3 : 397 · 727
4 : 2^2 · 2381107
5 : 5^3 · 7 · 359207
7 : 11027 · 31040117
15 : 13337 · 706661 · 51076716238627
19 : 38231 · 14932493857679888742000509
For n=15 & 19 same explanation as R15 k=47

R36 k=1555:
n-value : factors
1 : 11 · 727
2 : 31 · 37 · 251
3 : 67 · 154691
4 : 37 · 127 · 271 · 293
7 : 4943 · 3521755879
9 : 59 · 382386761790283
For n=7 & 9 same explanation as R15 k=47


The prime factors for n=12, n=15, and n=7 respectively make it clear to me that these k-values should all yield primes at some point so you are correct to include them as remaining.

The higher-math folks may be able to chime in and answer why there are an abnormally large # of k's that are perfect squares that end up remaining even though they don't have known algebraic factors for most bases. IMHO, it's because there ARE algebraic factors for a subset of the universe of n-values on them but not for all of the n-values. Hence they are frequently lower weight than the other k's but NOT zero weight and so should eventually yield a prime.
In fact, (k*b^n+1)/gcd(k+1,b-1) has algebra factors if and only if k*b^n is either perfect odd power or of the form 4*m^4, and (k*b^n-1)/gcd(k-1,b-1) has algebra factors if and only if k*b^n is perfect power, thus forms like (257*33^n-1)/32 cannot have algebra factors, since there is no n such that 257*33^n is perfect power (after all, the exponent of the prime 257 in the prime factorization of 257*33^n is 1 for all n, but any prime factor in the prime factorization of a perfect power cannot have exponent 1).

Last fiddled with by sweety439 on 2020-07-07 at 18:09
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Old 2020-07-07, 19:27   #869
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Quote:
Originally Posted by sweety439 View Post
(101*73^2146-1)/4 is (probable) prime!!!

R73 has only k=79 remain, reserve it to n=10K
(79*73^9339-1)/6 is (probable) prime!!!

R73 is proven!!!
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