Sierpinski's Problem to One Million
Joseph McLean
Sierpinski's
Problem is to remove as many odd k from consideration as Sierpinski numbers by
finding primes of the form k.2n +1, if necessary by pushing n to
very high limits, and as a by-product, locating some very large primes in the
process. For convenience, I make the following definitions :
A
Keller prime is a prime of the form k.2n +1, where k is odd, and for
which none of the numbers k.2m +1 for 0 < m < n is prime.
The
Keller prime for k, k being odd, is the smallest prime of the form k.2n
+1, with n > 0.
I
have previously considered all odd k between 78558 and
105. Of these, only 3 values of k still do not have a known
Keller prime.
As
a wholesale extension, I will now consider all odd k between 1 and 106.
Again, I only consider values of k for which there is no prime of the form k.2n
+1 for n £ 1000. A detailed breakdown
of Keller prime counts for n £ 1000 is available on
request. The counts of surviving k values, broken down into ranges of x.105
to (x+1).105 are as follows :
|
x |
k |
|
0 |
247 |
|
1 |
269 |
|
2 |
271 |
|
3 |
298 |
|
4 |
298 |
|
5 |
310 |
|
6 |
302 |
|
7 |
318 |
|
8 |
303 |
|
9 |
315 |
|
Total |
2931 |
The Nash weight of each of these was calculated to give some idea of the relative difficulty of searching. Additionally, given the nature of the values involved, this is a convenient method of obtaining values with Nash weight less than 100. These are useful when searching for very large primes. Nash weights are split as follows :
|
x |
w
= 0 |
<100 |
<200 |
<300 |
<400 |
<500 |
<600 |
<700 |
<800 |
<900 |
<1000 |
others |
totals |
|
0 |
1 |
11 |
40 |
58 |
49 |
41 |
21 |
11 |
7 |
6 |
1 |
1 |
247 |
|
1 |
0 |
6 |
40 |
76 |
51 |
44 |
27 |
13 |
7 |
3 |
2 |
0 |
269 |
|
2 |
2 |
10 |
52 |
65 |
55 |
38 |
23 |
8 |
10 |
4 |
2 |
2 |
271 |
|
3 |
2 |
10 |
49 |
85 |
64 |
35 |
27 |
12 |
8 |
1 |
2 |
3 |
298 |
|
4 |
1 |
6 |
46 |
78 |
70 |
45 |
27 |
14 |
7 |
2 |
2 |
0 |
298 |
|
5 |
1 |
10 |
46 |
81 |
61 |
54 |
25 |
20 |
5 |
4 |
2 |
1 |
310 |
|
6 |
1 |
7 |
48 |
59 |
73 |
51 |
28 |
18 |
9 |
4 |
3 |
1 |
302 |
|
7 |
0 |
7 |
46 |
78 |
69 |
50 |
29 |
19 |
9 |
3 |
3 |
5 |
318 |
|
8 |
0 |
6 |
48 |
65 |
74 |
46 |
28 |
16 |
10 |
5 |
4 |
1 |
303 |
|
9 |
3 |
5 |
51 |
75 |
63 |
47 |
27 |
21 |
12 |
5 |
3 |
3 |
315 |
|
tot |
11 |
78 |
466 |
720 |
629 |
451 |
262 |
152 |
84 |
37 |
24 |
17 |
2931 |
A
Nash weight of zero indicates a Sierpinski number. These, with covering sets,
are :
78557 - {3, 5, 7, 13, 19, 37, 73}
271129
- {3, 5, 7, 13, 17, 241}
271577
- {3, 5, 7, 13, 17, 241}
322523
- {3, 5, 7, 13, 37, 73, 109}
327739
- {3, 5, 7, 13, 17, 97, 257}
482719
- {3, 5, 7, 13, 17, 241}
575041 - {3, 5, 7, 13, 17, 241}
603713 - {3, 5, 7, 13, 17, 241}
903983 - {3, 5, 7, 13, 17, 241}
934909 - {3, 5, 7, 13, 19, 73, 109}
965431 - {3, 5, 7, 13, 17, 241}
For
each of the k with a Nash weight greater than zero, the search limit was
initially pushed up to 4000. This removes more than half of the values,
leaving, for each subrange :
|
x |
k |
|
0 |
128 |
|
1 |
134 |
|
2 |
149 |
|
3 |
146 |
|
4 |
133 |
|
5 |
151 |
|
6 |
142 |
|
7 |
138 |
|
8 |
148 |
|
9 |
144 |
|
Total |
1413 |
These
counts include Sierpinski numbers.
For
each of these k, the search limit was then pushed up to 10000, leaving, by
subrange :
|
x |
k |
|
0 |
88 |
|
1 |
85 |
|
2 |
102 |
|
3 |
96 |
|
4 |
90 |
|
5 |
95 |
|
6 |
94 |
|
7 |
85 |
|
8 |
100 |
|
9 |
93 |
|
Total |
928 |
The
remaining k are then treated individually with no common search limit. If a
Keller prime is found, the search will generally be halted immediately, unless
the Nash weight is low. In this case, the search for large Robinson primes
takes over, as explained elsewhere.
In
2002, a distributed project known as Seventeen or Bust (SB) was set up by Louis
Helm, David Norris & Michael Garrison to tackle the then 17 gaps in the
table for k < 78557. In 2010, after having found 11 large Keller primes, the
original project merged with Primegrid, a much more comprehensive project first
set up by Rytis Slatkevičius, and which uses the distributed software
BOINC. With Primegrid's help, another prime was found for k = 10223 in 2016.
In
2003, the Prime Sierpinski Project (PSP) was set up by Harsh Aggarwal in order
to find primes for the then 29 gaps between 78557 & 271129 for which k is
itself prime. This closed 18 gaps before merging with Primegrid.
The
three projects are now combined within Primegrid to search all gaps up to
271129. There are currently 5 values in SB remaining with k < 78557, 9
remaining in PSP. The values 22699 and 67607 are shared between these. In
addition, another 8 non-prime values of 78557 < k < 271129 are remaining,
the most recent value removed being 168451 in 2017 (see reference),
193997 in 2018 (see
reference), 99739 in 2019 (see reference) &
202705 in 2021 (see
reference).
For
a comprehensive view of the recent searches for k < 271129, see Wilfrid
Keller's ProthSearch page.
The
following is a list of all k that have no Keller prime up to n = 10000,
together with their Keller primes if known. Sierpinski numbers are marked with
asterisks. An up-to-date list of factors is
available in text form.
|
3061 33288 |
32161 43796 |
54767 1337287 |
78181 22024 |
|
4847 3321063 |
33661 7031232 |
55459 |
78557 ***** |
|
5297 50011 |
34711 10464 |
59569 390454 |
79309 |
|
5359 5054502 |
34999 462058 |
60443 95901 |
79817 |
|
5897 22619 |
36983 38573 |
60541 176340 |
80839 15030 |
|
7013 126113 |
37561 16604 |
62093 18353 |
81269 12979 |
|
7651 25368 |
39079 26506 |
62761 15064 |
84409 38070 |
|
8423 55157 |
39781 176088 |
63017 53195 |
85013 699333 |
|
10223 31172165 |
40547 12983 |
64007 26015 |
85711 12696 |
|
13787 53135 |
44131 995972 |
65567 1013803 |
86701 17768 |
|
14027 40639 |
44903 17913 |
67193 16249 |
86747 42051 |
|
16817 42155 |
46157 698207 |
67607 |
86869 11542 |
|
18107 21279 |
46187 104907 |
67759 10402 |
87743 212565 |
|
19249 13018586 |
46471 96640 |
69107 16599 |
89059 33834 |
|
20851 10672 |
47897 61871 |
69109 1157446 |
89225 92067 |
|
21181 |
48833 167897 |
71417 26807 |
90527 9162167 |
|
22699 |
49219 16102 |
71671 28884 |
91549 |
|
24737 |
50693 32753 |
74191 20340 |
93617 17587 |
|
25819 111842 |
51917 18031 |
74269 167546 |
94373 3206717 |
|
27653 9167433 |
53941 36944 |
75841 31220 |
98431 15880 |
|
27923 158625 |
54001 16652 |
76759 17446 |
98749 1045226 |
|
28433 7830457 |
54739 14282 |
77899 43194 |
99739 14019102 |
|
101869 77002 |
132439 23158 |
156889 65082 |
181639 27042 |
|
101911 10560 |
134131 19248 |
159503 540945 |
181921 148432 |
|
102259 19070 |
135887 32319 |
161041 7107964 |
182749 14890 |
|
102263 52853 |
137269 21958 |
161509 17154 |
183091 43984 |
|
105857 49155 |
137401 40956 |
161957 727995 |
183347 116399 |
|
106853 50061 |
138199 74670 |
163187 |
184609 23130 |
|
107929 1007898 |
139201 14752 |
163463 10069 |
185449 435402 |
|
110941 12340 |
142099 70802 |
165499 79638 |
185993 164613 |
|
112097 17539 |
145609 23798 |
167551 26092 |
187681 573816 |
|
114751 11064 |
146761 56816 |
167957 417463 |
190999 54478 |
|
115561 91548 |
147143 16973 |
168451 19375200 |
191537 34067 |
|
117557 51511 |
147391 120616 |
168587 545971 |
193801 26344 |
|
118069 27078 |
147559 2562218 |
169639 31018 |
193997 11452891 |
|
118081 145836 |
149183 1666957 |
170399 17995 |
196307 50267 |
|
118249 80422 |
151945 62876 |
172127 448743 |
197753 73745 |
|
122149 578806 |
152183 20333 |
172157 71995 |
198113 267005 |
|
123287 2538167 |
152267 |
173633 16177 |
198647 178863 |
|
128239 88330 |
153169
30478 |
173933 340181 |
198677 2950515 |
|
128449 109130 |
153263 21309 |
177421 69880 |
199037 101723 |
|
128789 31049 |
154801 1305084 |
177803 28653 |
|
|
130337 11563 |
155357 79679 |
179581 117980 |
|
|
131179 |
156511 |
179791
331740 |
|
|
200749 |
225679 620678 |
260251 14556 |
282049 21634 |
|
201407 20247 |
225931 |
261203 354561 |
283741 32592 |
|
202705 21320516 |
227723 |
261917 704227 |
284843 13517 |
|
203749 13014 |
227753 91397 |
263329 406934 |
285473 530921 |
|
203761 384628 |
229673 |
263927 639599 |
285601 |
|
203941 23004 |
231797 66503 |
264451 11940 |
286037 |
|
206821 22868 |
235351 13048 |
264977 26395 |
286051 13856 |
|
208381 463068 |
235607 60635 |
265711 4858008 |
286579 33842 |
|
209611 |
237019 |
267893 12237 |
287003 23169 |
|
210917 16251 |
237413 267149 |
268021 27844 |
287393 |
|
211195 3224974 |
238411 |
269041 10668 |
287509 17362 |
|
212671 15816 |
239063 21761 |
269153 12377 |
287899 223886 |
|
214519 1929114 |
240211 93184 |
270557 111807 |
288151 73752 |
|
214583 25225 |
240713 11497 |
271129 ***** |
289171 |
|
216751 903792 |
241489 1365062 |
271577 ***** |
289939 48286 |
|
218447 12667 |
244609 12050 |
272341 |
290281 61872 |
|
219259 1300450 |
246053 44673 |
273631 30340 |
291979 65182 |
|
220639 29650 |
247099 484190 |
273679 1052058 |
293723 16517 |
|
222113 |
250163 198453 |
274699 |
294181 |
|
222169 18338 |
250463 1316921 |
277217 72227 |
294241 184040 |
|
222361 2854840 |
252181 149684 |
278209 25518 |
297257 28211 |
|
222379 12842 |
255811 140148 |
278837 10031 |
297317 54755 |
|
223127 50959 |
256787 11455 |
278843 51801 |
297617 60187 |
|
224027 273967 |
258317 5450519 |
279361 1613712 |
299239 98714 |
|
224657 36391 |
258541 25308 |
279767 |
|
|
225077 20047 |
259733 29973 |
281543 440853 |
|
|
300649 23846 |
328703 17949 |
356599 15234 |
376879 13946 |
|
300661 22084 |
329221 19508 |
356809 35922 |
381811 17124 |
|
302701 71716 |
331883 119945 |
356971 156572 |
381991 14224 |
|
303001 12868 |
332093 18349 |
357017 332367 |
382247 |
|
303197 29047 |
334519 10902 |
357271 |
382849 15598 |
|
305063 |
335299 78574 |
359551 111552 |
383717 325171 |
|
305147 1030527 |
335453 28633 |
359747 11235 |
383843 87973 |
|
305363 22385 |
335957 30331 |
359933 |
385313 142989 |
|
305807 12511 |
336143 49353 |
360331 |
385789 15658 |
|
306893 117721 |
336923 12649 |
362281 11172 |
388561 16956 |
|
307877 27435 |
337111 229176 |
362881 |
389401 171604 |
|
308237 48491 |
337651 31492 |
363917 |
389581 75348 |
|
308423 395337 |
340171 31976 |
364969 98546 |
390533 13761 |
|
310339 |
340441 |
365033 23121 |
391681 52736 |
|
311267 89215 |
340471 128256 |
365221 |
392033 |
|
311573 |
340759 |
365867 |
393091 16204 |
|
312121 1109856 |
348169 17166 |
366467 88163 |
393497 255283 |
|
312469 64970 |
348353 19585 |
366953 |
394429 16626 |
|
312863 293881 |
348587 99519 |
368299 |
396203 480729 |
|
315409 12294 |
350689 250850 |
369497 183079 |
397309 296070 |
|
322523 ***** |
351167 |
371027 103775 |
398089 11794 |
|
324169 15802 |
352217 26827 |
374093 36925 |
398597 34779 |
|
327679 24046 |
353239 12906 |
376141 87980 |
399481 48792 |
|
327739 ***** |
354203 95549 |
376463 13197 |
399617 340955 |
|
400613 |
427339 78174 |
450589 67886 |
475817 14767 |
|
401131 37932 |
427517 82787 |
451051 29360 |
479783 |
|
401371 62400 |
428387 21123 |
451351 |
480863 25785 |
|
402407 34875 |
428657 720223 |
451975 11650 |
481727 883059 |
|
404177 10511 |
430303 87694 |
452003 48185 |
482719 ***** |
|
408379 27854 |
430727 88875 |
452119 |
483233 48389 |
|
409279 27662 |
432257 12203 |
452131 18836 |
483661 14088 |
|
411953 78929 |
432983 259453 |
452371 190272 |
484763 |
|
412081 145336 |
433261 11652 |
454159 24158 |
487681 19180 |
|
412367 25779 |
433457 |
456211 325780 |
487819 27486 |
|
412591 |
437933 51817 |
456347 50039 |
488341 466940 |
|
413537 92055 |
438541 60832 |
457217 |
488581 27344 |
|
413873 32701 |
438773 11357 |
459893 11377 |
488641 13804 |
|
415313 26481 |
440837 24831 |
462829 |
488843 18493 |
|
415427 50571 |
441923 774725 |
464929 59426 |
489977 49299 |
|
415523 11393 |
443701 30008 |
465407 159915 |
491147 |
|
418487 10771 |
445373 16377 |
467543 21185 |
492953 260957 |
|
418591 |
446509 557118 |
467963 132769 |
495979 480286 |
|
419093 42161 |
446633 107905 |
469073 104969 |
498107 22483 |
|
420113 524009 |
447061 1206128 |
470477 17471 |
498781 557856 |
|
422491 85556 |
447079 50514 |
470693 35533 |
499561 |
|
423083 23817 |
447271 10856 |
473567 |
|
|
425149 80674 |
450139 54914 |
475549 12194 |
|
|
501107 |
529759 10922 |
554659 18470 |
574907 14395 |
|
501143 113609 |
532703 |
556667 86543 |
575041 ***** |
|
502613 |
533213 11045 |
556697 13371 |
575539 431950 |
|
504199 37794 |
536329 27362 |
557693 117549 |
575791 10760 |
|
506749 574746 |
536779 67590 |
559051 33872 |
578689 66070 |
|
507743 30133 |
536839 |
559549 25158 |
583189 |
|
508217 16343 |
538313 66301 |
561769 |
583441 14328 |
|
509101 14952 |
538943 |
562487 18603 |
583561 10724 |
|
510893 556521 |
540977 188027 |
564511 82868 |
583939 12290 |
|
515161 17832 |
541877 82575 |
565681 14104 |
587417 10983 |
|
515357 |
541939 16522 |
566011 16464 |
588317 |
|
515611 15256 |
542093 11177 |
566569 |
589021 |
|
517651 204528 |
544433 16805 |
567587 18935 |
590033 |
|
517883 14533 |
545401 |
568067 107603 |
591323 13477 |
|
517913 |
545959 30014 |
569581 11572 |
593417 43043 |
|
518671 |
546529 264498 |
570601 65040 |
593689 12030 |
|
522001 39244 |
548033 |
570923 207189 |
594151 22264 |
|
523547 25203 |
548869 304442 |
571969 10262 |
596959 22294 |
|
523669 202714 |
550429 12894 |
572029 52890 |
597211 26912 |
|
524663 423169 |
551071 25260 |
572213 66409 |
597323 21949 |
|
525409 98942 |
552203 57841 |
572491 37860 |
599003 |
|
527791 51192 |
552593 24517 |
572507 133971 |
599011 |
|
528139 12050 |
553159 |
573271 42820 |
599513 |
|
529157 22171 |
554573 305373 |
574061 13317 |
|
|
602537 19939 |
621953 151469 |
652291 27960 |
678739 16638 |
|
603593 12221 |
624139 53814 |
653063 899301 |
680597 10399 |
|
603713 ***** |
624511 962636 |
653693 93457 |
680851 741248 |
|
603767 |
626303 |
654083 217533 |
682141 36628 |
|
604189 500578 |
626419 211622 |
655367 16171 |
682667 15955 |
|
606199 |
629089 11966 |
656123 |
683087 18563 |
|
607801 190960 |
629149 52250 |
656753 |
684977 14543 |
|
608179 32854 |
630121 |
659237 11295 |
686479 372062 |
|
609227 |
632663 |
662345 10099 |
687347 12087 |
|
609769 879034 |
633197 73171 |
662899 42658 |
688819 36538 |
|
610097 330651 |
633407 16627 |
663827 65447 |
689281 17724 |
|
612263 112221 |
633481 |
665101 10116 |
692063 20293 |
|
612773 51941 |
633841 26868 |
665423 566441 |
692431 56560 |
|
614617 21150 |
634441 14928 |
667861 |
693769 247742 |
|
615151 800316 |
637501 20440 |
670309 520410 |
694879 169326 |
|
616367 30915 |
639827 32291 |
670969 36346 |
694891 33428 |
|
618379 143758 |
644333
23401 |
672481 10896 |
694973 |
|
618889 15574 |
646411 23372 |
673009 60186 |
695659 10378 |
|
618893 11845 |
646427 67667 |
673667 11963 |
695911 28156 |
|
619013 849281 |
646757 20963 |
674477 |
696323 15057 |
|
619399 150478 |
646937 23223 |
675559 23858 |
696661 134472 |
|
619403 76141 |
648751 |
678047 74863 |
697831 12476 |
|
620063 15825 |
651661 17268 |
678109 54994 |
|
|
621437 16455 |
652067 12155 |
678173 |
|
|
700141 11572 |
724351 675612 |
754939 10062 |
777559 |
|
700339 21630 |
725483 37421 |
756139 28594 |
778021 55132 |
|
700637 14795 |
730831 13272 |
757343 |
783019 235798 |
|
701123 57057 |
731041 21884 |
758243 161705 |
784159 11310 |
|
701357
532979 |
733193 22265 |
759653 154085 |
785209 37266 |
|
702707 |
734147 1047447 |
760283 13729 |
787489 19634 |
|
703643 |
734753 77481 |
760583 |
787981 36844 |
|
706627 18322 |
735287 10883 |
761749 716354 |
791273 14557 |
|
708263 20673 |
736249 |
762227 325531 |
791491 17468 |
|
709187 30675 |
736999 10026 |
762769 188586 |
791867 13387 |
|
710603 12513 |
741301 14780 |
766531 |
793051 14984 |
|
711721 13116 |
743357 860491 |
766801 |
793561 19576 |
|
711833 |
745337 27295 |
769343 1230661 |
793817 106719 |
|
714563 |
745673 45969 |
769861 41364 |
794867 |
|
715531 178144 |
747379 49742 |
770867 14975 |
795983 |
|
717001 58936 |
749447 1036639 |
770899 25346 |
796493 17745 |
|
718463 199345 |
749971 843268 |
771977 21339 |
797131 39472 |
|
718849 |
750083 684961 |
772403 13277 |
797903 26677 |
|
719611 79296 |
751103 26481 |
772969 26754 |
799951 13500 |
|
720121 86960 |
751613 31745 |
774977 261235 |
|
|
721141 |
751999 |
775969 181806 |
|
|
721969 250898 |
753047 16115 |
777503 22493 |
|
|
801349 27426 |
822083 670761 |
846857 453343 |
877583 17521 |
|
801923 690713 |
822631 11424 |
850337 11307 |
877997 21395 |
|
802367 47495 |
823969 |
851537 18295 |
878029 |
|
802493 24341 |
826201 |
852019 |
879049 |
|
802613 |
829013 457421 |
853817 23795 |
879497 |
|
804019 25462 |
829643 |
855017 12863 |
879919 23650 |
|
804173 22961 |
829681 13840 |
855949 16878 |
881537 |
|
804541 50676 |
831571 10492 |
856043 |
884723 |
|
804691 64628 |
833179 387618 |
856369 15350 |
885077 15891 |
|
804863 18045 |
835667 18907 |
856607 185727 |
885233 13625 |
|
805321 40024 |
835733 64485 |
858079 |
886699 14346 |
|
807203 11769 |
836161 69692 |
858527 14187 |
887153 |
|
808981 34548 |
836209 225602 |
859523 14629 |
888001 33892 |
|
810949 98090 |
838081 51712 |
861083 40441 |
889187 52975 |
|
811247 15635 |
838157 12847 |
861437 22007 |
889727 391475 |
|
812717 29091 |
838441 19248 |
863431 98416 |
890333 18561 |
|
812881 663152 |
840811 205948 |
865549 42098 |
892247 219359 |
|
813707 10567 |
841757 56951 |
867271 |
892249 141518 |
|
815491 128160 |
842393 44681 |
867443 57845 |
894353
26713 |
|
816353 20041 |
843079 |
868531 27712 |
894409 |
|
817021 22584 |
843317 38771 |
870061 |
894827 |
|
817403 43897 |
843811 39228 |
872119 |
895579 |
|
818551 112120 |
844703 15445 |
873227 10243 |
895823 495837 |
|
819437 |
846347 |
875447 10027 |
896851 31308 |
|
821071 12656 |
846737 25751 |
876529 43242 |
899449 |
|
900317 29263 |
928997 145355 |
957977 |
974483 69065 |
|
901067 |
930079 61386 |
958201 18868 |
976523 64801 |
|
903211 26344 |
931517 65647 |
959689 17778 |
977521 101792 |
|
903983 ***** |
933511 36760 |
959929 141906 |
979039 |
|
904081 11616 |
934909 ***** |
959971 44128 |
981623 33353 |
|
904489 |
936083 16009 |
960139 27378 |
982531 41652 |
|
907021 33968 |
937639 45566 |
960301 430616 |
982547 73643 |
|
908183 10809 |
939259 10298 |
961099 |
983027 |
|
909473 13569 |
941207 19535 |
961313 155421 |
983663 44377 |
|
910733 200233 |
941741 51345 |
961621 35996 |
984185 10165 |
|
911123 479981 |
941993 31745 |
963227 196403 |
988741 23532 |
|
911791 129892 |
942227 215687 |
963661 35212 |
989147 635919 |
|
912721 47292 |
942233 32421 |
964549 82462 |
989567 31683 |
|
912889 10342 |
943373 304161 |
964673 |
990421 80560 |
|
913847 11127 |
943981 20672 |
965431 ***** |
990721 22188 |
|
918001 11616 |
944011 372216 |
965689 23782 |
991037 10423 |
|
919291 14756 |
945499 42134 |
968491 |
991951 40688 |
|
922081 65840 |
947963 10225 |
969533 75885 |
992731 731740 |
|
922463 141321 |
951233 14673 |
969769 33270 |
993767 186619 |
|
923177 611483 |
951593 159929 |
970457 11679 |
996811 16788 |
|
924683 421877 |
951961 39888 |
971071 19716 |
997699 18902 |
|
924773 13121 |
953597 62719 |
971389 |
|
|
925907 |
954211 28976 |
972443 17033 |
|
|
927181 39656 |
956749 95966 |
972739 |
|
Keller
primes in the above tables are credited to the following (in no particular
order):
Jeff Young
Duncan Buell & Jeff Young
Joe McLean*
Jim Fougeron*
Jim McElhatton*
Dirk Augustin*
Dan Morenus*
Ray Ballinger @
Sander Hoogendoorn*
Kimmo Herranen*
Brian Schroeder*
Contributors from the Prime Sierpinski Project,
administrated by Harsh Aggarwal
Nestor Melo @
Daval
Phil Carmody @
Pavlos Saridis @
Pavel Chernishev
Contributors from the Seventeen or Bust Project,
administrated by Louis Helm, David Norris & Michael Garrison
Contributors from the Proth Prime Search project using
Primegrid
The projects listed above cover particular subsets of the entire range under consideration. For any significant dent in the remainder, I have to thank contributors to my own co-ordinated search, who are marked with an asterisk. An ampersand indicates people whose help was invaluable in the earlier days of the search.
The first Keller primes found during this search and with an exponent in excess of one million are: 305147*21030527+1 & 312121*21109856+1 obtained by Dan Morenus on (or slightly before) 03/12/2017 & 04/01/2018 respectively.
The number of survivors of complete searches to n = 100000, 200000, 300000, 400000 & 500000, including Sierpinski numbers, is are follows:
|
x |
k |
k |
k |
k |
k |
|
0 |
37 |
29 |
28 |
27 |
26 |
|
1 |
33 |
24 |
23 |
21 |
18 |
|
2 |
48 |
43 |
40 |
38 |
34 |
|
3 |
40 |
31 |
26 |
22 |
21 |
|
4 |
33 |
27 |
25 |
24 |
22 |
|
5 |
36 |
31 |
27 |
25 |
23 |
|
6 |
36 |
29 |
26 |
24 |
24 |
|
7 |
35 |
28 |
25 |
24 |
24 |
|
8 |
38 |
35 |
32 |
30 |
27 |
|
9 |
33 |
24 |
22 |
20 |
17 |
|
Total |
369 |
301 |
274 |
255 |
236 |
The
above tables can be summarised as follows, where the counts are of k for which
no Keller prime is known, including and excluding Sierpinski numbers.
|
x |
k |
k exc |
|
0 |
9 |
8 |
|
1 |
4 |
4 |
|
2 |
18 |
16 |
|
3 |
19 |
17 |
|
4 |
14 |
13 |
|
5 |
21 |
20 |
|
6 |
15 |
14 |
|
7 |
15 |
15 |
|
8 |
23 |
23 |
|
9 |
14 |
11 |
|
Total |
152 |
141 |
The
extended Sierpinski search to find Keller primes for the remaining values of k
> 78557 not covered by any other project has an on-line
status report updated regularly.
The
number of outstanding k values left at each significant stage is as follows :
|
search
limit (on n) |
outstanding
k values |
|
1000 |
2931 |
|
2000 |
2002 |
|
3000 |
1645 |
|
4000 |
1413 |
|
5000 |
1282 |
|
10000 |
928 |
|
20000 |
701 |
|
50000 |
489 |
|
100000 |
368 |
|
200000 |
299 |
|
300000 |
272 |
|
500000 |
234 |
|
1000000 |
192 |
The
figures up to 1000000 have now been confirmed complete. Above this limit, the
values are not exhaustive and so may drop.
Last
updated: 24/08/2023