Searching for large Sierpinski numbers
In an earlier article, I introduced a method for generating numbers which have a good chance of having a Nash weight of less than 100. A sideeffect of this is of course the locating of Sierpinski numbers themselves, which have a Nash weight of zero. In fact the method may be modified specifically to find Sierpinski numbers, by ignoring k as soon as any value of n between 1 and 1000 survives the trial division, although in practice it is convenient to preserve the possibility of one or two exceptions. It takes only a few minutes to run this for all k up to one million, and it produces a list of 11 survivors, all of which, when run through the Nash sieve, are identified as Sierpinski numbers.
It is important to point out that for each k, the exponent n must be run to 1000. For example, if the modified algorithm is run for odd k between 70000 and 100000, and n only up to 100, then instead of leaving one possible candidate, there are 16 to be considered. Some of these, to be fair, have very low Nash weights, but only one (78557) has zero Nash weight.
It is possible that additional Sierpinski numbers are not identified because either
a) the Sierpinski number is divisible by 3, 5 or 7
or
b) a member of the covering set is greater than 6400 (the sieve limit).
As will become apparent, the dangers posed by condition a) are removed by the nature of certain relationships governing Sierpinski numbers. Also, there is a margin of error in the search algorithm that will cope with covering sets with one or two large primes.
The following list gives all Sierpinski numbers found using this method for k < 10^{7}. As a comparison, the algorithm was run for all k < 5.10^{6} without the restriction that k is not divisible by 3, 5, or 7, and no other Sierpinski numbers were located. I am therefore confident that the number of Sierpinski numbers not identified in the range of k searched is very small and may, in fact, be zero.
The covering set for a Sierpinski number k is a subset of the divisors of 2^{s}  1 for some minimal value s. This number is given in the second column and is called the Sierpinski modulus for k. Covering sets themselves are lettercoded in the fourth column, as follows :
code 
modulus 
covering set 
A 
24 
{3, 5, 7, 13, 17, 241} 
B 
36 
{3, 5, 7, 13, 19, 37, 73} 
C 
48 
{3, 5, 7, 13, 17, 97, 257} 
D 
36 
{3, 5, 7, 13, 19, 73, 109} 
E 
36 
{3, 5, 7, 13, 19, 37, 109} 
F 
36 
{3, 5, 7, 13, 37, 73, 109} 
G 
72 
{3, 5, 7, 13, 17, 19, 109, 433} 
It is obvious that if k is a Sierpinski number and P is the product of the primes in the covering set for k, then all the numbers generated by the iteration k ® (2k+P) mod 2P are also Sierpinski numbers with the same covering set. This iterative process repeats with period equal to the Sierpinski modulus s of k and is known as the Keller iteration. The sequence of s repeating numbers so generated is called a Keller cycle. In the fifth column below we attempt to indicate this using the following code: proceeding sequentially, if a value of k has not been previously identified as being a member of a particular cycle, then it is given the code N.0 where N is a number that starts at 1 and is incremented every time a new cycle is identified. The Keller cycle is then generated and a member of the cycle is labelled N.i if it is the ith iteration of the cycle. We proceed in this manner until all the values of k are labelled. Note that N.0 = N.s, where s is the Sierpinski modulus.
For interest, the third column contains either the word "prime" if k is prime, or the prime factors of k otherwise. It is hoped that a pattern may emerge regarding divisibility properties.
The Sierpinski numbers are :
k 
modulus 
factorisation 
covering set 
label 
78557 
36 
17.4621 
B 
1.0 
271129 
24 
prime 
A 
2.0 
271577 
24 
59.4603 
A 
3.0 
322523 
36 
prime 
F 
4.0 
327739 
48 
prime 
C 
5.0 
482719 
24 
prime 
A 
3.17 
575041 
24 
29.79.251 
A 
2.6 
603713 
24 
11.71.773 
A 
3.6 
903983 
24 
149.6067 
A 
2.13 
934909 
36 
prime 
D 
6.0 
965431 
24 
227.4253 
A 
2.18 
1259779 
36 
23.54773 
D 
6.24 
1290677 
36 
137.9421 
E 
7.0 
1518781 
24 
11.138071 
A 
3.15 
1624097 
24 
887.1831 
A 
2.11 
1639459 
24 
prime 
A 
2.16 
1777613 
72 
29.61297 
G 
8.0 
2131043 
24 
prime 
A 
2.21 
2131099 
24 
prime 
A 
3.21 
2191531 
36 
103.21277 
B 
1.31 
2510177 
36 
167.15031 
B 
9.0 
2541601 
24 
43.59107 
A 
3.11 
2576089 
36 
prime 
B 
9.21 
2931767 
24 
859.3413 
A 
2.23 
2931991 
24 
37.109.727 
A 
3.23 
3083723 
24 
443.6961 
A 
2.5 
3098059 
24 
prime 
A 
3.5 
3555593 
24 
23.154591 
A 
3.14 
3608251 
24 
prime 
A 
2.10 
4067003 
24 
37.109919 
A 
3.10 
4095859 
36 
41.283.353 
E 
7.29 
4573999 
24 
prime 
A 
3.13 
5455789 
48 
59.89.1039 
C 
10.0 
5841947 
36 
271.21557 
E 
11.0 
6134663 
24 
19.322877 
A 
2.1 
6135559 
24 
29.211571 
A 
3.1 
6557843 
24 
37.177239 
A 
3.18 
6676921 
24 
197.33893 
A 
2.2 
6678713 
24 
prime 
A 
3.2 
6742487 
24 
1327.5081 
A 
2.7 
6799831 
24 
prime 
A 
3.7 
6828631 
36 
23.337.881 
D 
12.0 
7134623 
36 
23^{2}.13487 
B 
9.14 
7158107 
36 
11.191.3407 
E 
7.20 
7400371 
24 
11.101.6661 
A 
2.14 
7523267 
24 
29.59.4397 
A 
2.19 
7523281 
24 
prime 
A 
3.19 
7696009 
36 
83.92723 
B 
1.25 
7761437 
24 
prime 
A 
2.3 
7765021 
24 
11.113.6247 
A 
3.3 
7892569 
24 
31.47.5417 
A 
2.8 
8007257 
24 
503.15919 
A 
3.8 
8184977 
36 
prime 
B 
13.0 
8629967 
24 
41.210487 
A 
3.16 
8840599 
24 
prime 
A 
2.12 
8871323 
24 
401.22123 
A 
2.17 
8879993 
48 
prime 
C 
10.33 
8959163 
36 
prime 
E 
14.0 
9043831 
48 
283.31957 
C 
5.42 
9044629 
36 
11^{2}.17.4397 
E 
7.13 
9208337 
24 
prime 
A 
2.15 
9252323 
36 
prime 
E 
11.14 
9454129 
24 
37.255517 
A 
2.20 
9454157 
24 
73.129509 
A 
3.20 
9854491 
24 
163.60457 
A 
2.22 
9854603 
24 
11^{2}.23.3541 
A 
3.22 
9930469 
24 
prime 
A 
2.4 
9933857 
36 
31^{2}.10337 
E 
7.24 
9937637 
24 
prime 
A 
3.4 
The counts for each covering set are as follows:
Total 
A 
B 
C 
D 
E 
F 
G 
69 
45 
7 
4 
3 
8 
1 
1 
In the above list, there are several obvious pairs, with differences, as follows :
271129 + 2^{6}.7 = 271577
2131043 + 2^{3}.7 = 2131099
2931767 + 2^{5}.7 = 2931991
6134663 + 2^{7}.7 = 6135559
6676921 + 2^{8}.7 = 6678713
7523267 + 2.7 = 7523281
7761437 + 2^{9}.7 = 7765021
9454129 + 2^{2}.7 = 9454157
9854491 + 2^{4}.7 = 9854603
where I have laid these out so that the striking similarity of the differences is apparent. Two other possible pairings, {8871323, 8879993} and {9043831, 9044629}, do not provide anything of interest, and the "closeness" of these pairs may be put down to coincidence. There is one other grouping of interest, that is, the last three. The two consecutive differences do not indicate anything of significance. However, the following may seem rather striking :
9930469 + 2^{10}.7 = 9937637
In other words, the difference 2^{m}.7 occurs once and only once for each m. It comes as no surprise to find that all these pairs have the same Sierpinski modulus and covering set, whereas the two other pairs mentioned do not. Additional investigation of differences between nonadjacent Sierpinski numbers reveals the following :
3083723 + 2^{11}.7 = 3098059
This effect is a direct result of the cyclic generating procedure, since if k and k' have the same covering set and we let x = k'  k, then the differences between successive iterations in the Keller cycle are just 2x, 4x, etc. In the above example, we may take the closest pairing, that is, k = 7523267 and k' = 7523281 as the starting values of the cycles.
Obviously, x = k'  k is even. The above example has x = 2*7. It would be interesting to find examples with x = 2*3 or 2*5, although at present no other pair of known Sierpinski numbers comes anywhere near as close as this.
We shall now continue the list to 2*10^{7}. It is obvious that the alternative iteration k ® (k+2P) provides an infinite sequence of Sierpinski numbers. However, these have identical properties to the associated base values and will be labelled with an asterisk. Applying the Keller iteration to any one of these generated values will recreate the Keller cycle for the base value. A Keller cycle is therefore uniquely defined by a base value N.0 and a covering set.
10192733 
24 
prime 
A 
2.9 
10275229 
36 
31.461.719 
B 
9.5 
10306187 
36 
prime 
E 
15.0 
10391933 
36 
prime 
B 
1.10 
10422109 
24 
47.221747 
A 
3.9 
10675607 
24 
1297.8231 
A 
3.12 
11000303 
36 
43.47.5443 
E 
15.22 
11201161 
36 
23.487007 
B 
13.31 
11206501 
48 
prime 
H 
16.0 
11455939 
24 
11.1041449 
A 
* 
11456387 
24 
prime 
A 
* 
11667529 
24 
prime 
A 
* 
11759851 
24 
661.17791 
A 
* 
11788523 
24 
prime 
A 
* 
11822359 
36 
prime 
D 
17.0 
12088793 
24 
prime 
A 
* 
12150241 
24 
521.23321 
A 
* 
12151397 
36 
149.81553 
B or D or J 
9.8 or 12.5 or 18.0 
12384413 
36 
337.36749 
B 
9.26 
12413281 
36 
17.191.3823 
F 
19.0 
12703591 
24 
149.85259 
A 
* 
12756019 
36 
prime 
B 
20.0 
12808907 
24 
19.23.29311 
A 
* 
12824269 
24 
67.277.691 
A 
* 
13065289 
36 
89.146801 
B 
13.13 
13085029 
36 
43.304303 
B 
1.13 
13315853 
24 
43.309671 
A 
* 
13315909 
24 
prime 
A 
* 
13726411 
24 
prime 
A 
* 
14116577 
24 
prime 
A 
* 
14116801 
24 
1873.7537 
A 
* 
14268533 
24 
23.41.15131 
A 
* 
14282869 
24 
397.35977 
A 
* 
14740403 
24 
prime 
A 
* 
14793061 
24 
prime 
A 
* 
15168739 
36 
3253.4663 
B 
19.20 
15251813 
24 
19.67.11981 
A 
* 
15273751 
48 
prime 
I 
21.0 
15285707 
36 
prime 
D 
12.13 
15598231 
36 
11^{2}.17.7583 
E 
11.7 
15758809 
24 
11.19.75401 
A 
* 
16010419 
36 
101.158519 
E 
15.5 
16391273 
36 
71.230863 
B 
19.9 
16625747 
36 
prime 
E 
7.8 
16907749 
36 
73.231613 
E 
11.25 
16921847 
36 
prime 
F 
18.13 
17220887 
36 
3709.4643 
D 
22.0 
17319473 
24 
prime 
A 
* 
17320369 
24 
11.1574579 
A 
* 
17742653 
24 
prime 
A 
* 
17861731 
24 
23.67^{2}.173 
A 
* 
17863523 
24 
601.29723 
A 
* 
17927297 
24 
prime 
A 
* 
17984641 
24 
prime 
A 
* 
18068693 
36 
prime 
E 
11.30 
18140153 
36 
prime 
B 
9.34 
18156631 
36 
prime 
B 
9.19 
18585181 
24 
prime 
A 
* 
18708077 
24 
2707.6911 
A 
* 
18708091 
24 
prime 
A 
* 
18946247 
24 
239.79273 
A 
* 
18949831 
24 
41.462191 
A 
* 
19077379 
24 
67.284737 
A 
* 
19192067 
24 
prime 
A 
* 
19428919 
36 
prime 
E 
15.13 
19436611 
36 
197.98663 
B 
1.23 
19558853 
36 
971.20143 
B 
13.34 
19814777 
24 
19.47.22189 
A 
* 
where we have the additional coded covering sets :
code 
modulus 
covering set 
H 
48 
{3, 5, 7, 17, 97, 241, 257} 
I 
48 
{3, 5, 7, 13, 17, 257, 673} 
J 
36 
{3, 5, 7, 13, 19, 73, 4033} 
The algorithm actually located one value, k = 13965257, which had a zero intermediate weighting but a nonzero Nash weight and is therefore not a Sierpinski number.
As has been indicated above, it is entirely possible for a particular Sierpinski number to have more than one covering set. We shall call such numbers "overlapping" Sierpinski numbers, or "overlaps". The set J is not a true covering set since 4033 = 37*109 is not a prime. However, it is included in order to label these overlaps conveniently.
For covering set A, we have 2P(A) = 11184810. The only two base values with this covering set that are less than this bound are 271129 and 271577. Since the search has now passed this value of 2P, all other Sierpinski numbers with this covering set can be generated from one of these two by using either the Keller iteration or the additive iteration. This idea stretches to all Sierpinski numbers, that is, for a given covering set and modulus, all Sierpinski numbers k such that k > 2P have an associated base value k' such that k' < 2P. Thus there is a finite number of base values, and therefore also of Keller cycles, for each finite covering set.
None of the above Sierpinski numbers is divisible by 13. However, we cannot assume that this is a universal property. The reasoning for this is as follows: the number k = 11206501 has covering set H, as can be seen above, which does not contain 13. The product P(H) is approximately 10^{10} and the Sierpinski modulus is 48. Applying the Keller iteration and ignoring any number greater that 2^{31} (which is the limit of the NashJobling implementation of the Nash sieve), we are left with the following :
iteration # 
Sierpinski number 
factorisation 
Sierpinski modulus 
10 
751375159 
19.prime 
48 
14 
1297920679 
11.31.317.12007 
24 
38 
701684269 
11.13.19.101.2557 
48 
42 
502866439 
prime 
48 
Thus 701684269 is a Sierpinski number which is divisible by 13. A bonus here is that this value is 20 times smaller than the expected theoretical value.
Iteration 14 above is unusual. Although the original covering set still covers this value of k, the prime 13 has the same effect as 97 and 257 combined. We shall return to this idea shortly.
The 22 Keller cycles identified with smallest values less than 2*10^{7} are divided amongst covering sets as follows :
A 
271129 (2) 
271577 (3) 


B 
78557 (1) 
2510177 (9) 
8184977 (13) 
12756019 (20) 
C 
327739 (5) 
5455789 (10) 


D 
934909 (6) 
6828631 (12) 
11822359 (17) 
17220887 (22) 
E 
1290677 (7) 
5841947 (11) 
8959163 (14) 
10306187 (15) 
F 
322523 (4) 
12413281 (19) 


G 
1777613 (8) 



H 
11206501 (16) 



I 
15273751 (21) 



J 
12151397 (18) 



where the values in brackets are the labels given to the Sierpinski number in the earlier lists.
There are 7 Sierpinski numbers that overlap between cycles with different covering sets. These are :
12151397  shared by cycles 9 (B) and 12 (D)
45181667  shared by cycles 13 (B) and 6 (D)
68468753  shared by cycles 1 (B) and 17 (D)
69169397  shared by cycles 21 (D) and 11 (E)
71307347  shared by cycles 1 (B) and 17 (D)
182479909  shared by cycles 12 (D) and 14 (E)
392581699  shared by cycles 6 (D) and 18 (F)
The four overlaps involving covering sets B and D are similar in that the primes 37 and 109 are directly interchangeable since they cover the same exponents, that is, have the same Nash congruence. Since these two primes both have exponent 36 to base 2, there is a 35 to 1 chance that a Sierpinski number covered by B or D is covered by both B and D. Any overlap between B and D is automatically covered also by J since 4033 = 37*109.
The two overlaps involving covering sets D and E are more complicated in that the subsets {3,5,73) of D and {3,5,37} of E are interchangeable but no smaller subsets are interchangeable. This is because the Nash congruence for 37 is 27 mod 36 and for 73 is 0 mod 9. Thus 73 covers everything covered by 37 but the reverse is not the case. However, 0, 9 and 18 mod 36 are covered by the primes 3 and 5 so the net effect of the triplets is the same. Similarly, with the overlap involving covering sets D and F, the subsets {5,19} and {5,37} are interchangeable since 19 covers both 4 and 22 mod 36 whereas 37 covers only 22 mod 36, but the prime 5 covers all numbers divisible by 4. In the event of an overlap between D and E or D and F, the set D is considered to be the better covering set since the exponent of the exceptional prime is lower in each case.
Other overlaps exist, for instance the one mentioned previously with respect to the search for a Sierpinski number divisible by 13, in which the overlap involves covering sets of different moduli. A complete picture of overlaps involving covering set A is missing since the product of primes is so small that it requires generated values to match those in Keller cycles of other covering sets and only base values where compared above.
Continuing the search to 5*10^{7} reveals 3 more covering sets, 7 more Keller cycles and 8 more overlaps between different Keller cycles. Ignoring those Sierpinski numbers with modulus 24 (as they are all generated values), the list continues :
20189993 
36 
967.20879 
E 
11.10 
20312899 
36 
43.472393 
B 
13.29 
20778931 
36 
71.292661 
B 
13.11 
21610427 
36 
prime 
B 
20.7 
21823667 
48 
937.23291 
C 
5.19 
22024609 
72 
1303.16093 
K 
23.0 
22047647 
36 
1481.14887 
B 
9.32 
23277113 
36 
79.294647 
D 
6.13 
23487497 
36 
11.2135227 
E 
15.8 
24885199 
48 
113.191.1153 
L 
24.0 
25614893 
36 
23.311.3581 
E 
14.24 
25763447 
48 
983.26209 
C 
10.15 
25912463 
36 
47.479.1151 
D 
6.33 
26471633 
36 
67^{2}.5897 
E 
7.34 
27160741 
36 
373.72817 
B 
9.11 
27862127 
36 
29.960763 
F 
25.0 
28410121 
36 
113.251417 
E 
7.11 
29024869 
36 
67.433207 
B 
9.29 
29095681 
36 
839.34679 
F 
25.27 
29949559 
48 
prime 
C 
5.32 
30375901 
36 
prime 
E 
11.23 
30423259 
36 
prime 
B 
20.28 
30666137 
36 
29.47.149.151 
E 
11.28 
31997717 
36 
prime 
B 
20.3 
32548519 
36 
23.53.26701 
B 
20.24 
32552687 
36 
67.289.1249 
E 
14.22 
32971909 
36 
311.106019 
D 
6.8 
33234767 
36 
prime 
B 
20.31 
33485483 
36 
prime 
B 
13.6 
33742939 
36 
283.119233 
E 
11.21 
34167691 
36 
929.36779 
B 
20.14 
34471877 
36 
11.1493.2099 
B 
13.16 
34629797 
36 
193.179429 
B 
1.16 
34636643 
36 
23.29.51929 
B 
13.22 
34689511 
36 
prime 
D 
12.28 
35430841 
36 
127.227.1229 
F 
4.21 
36029731 
72 
71.507461 
G 
26.0 
36120983 
36 
31.1165193 
B 
1.30 
36234799 
36 
3307.10957 
E 
14.31 
37158601 
48 
prime 
M 
27.0 
38206517 
36 
prime 
D 
12.17 
38222131 
36 
349.109519 
E 
15.27 
38257411 
36 
prime 
D 
22.19 
38592529 
36 
4937.7817 
B 
9.13 
38750753 
36 
2971.13043 
E 
14.4 
38942027 
36 
1049.37123 
E 
14.14 
39953689 
36 
17.2350217 
E 
11.33 
40118209 
36 
prime 
E 
7.17 
40343341 
36 
prime 
E 
14.33 
40511719 
36 
43.83.11351 
E 
15.33 
41134369 
48 
prime 
L 
28.0 
41403227 
36 
3533.11719 
B 
20.35 
42609587 
36 
prime 
B 
20.19 
43441313 
36 
211.205883 
E 
11.18 
43925747 
36 
43.107.9547 
F 
25.4 
44091199 
36 
prime 
E 
14.11 
44103533 
36 
4751.9283 
B 
9.18 
44743523 
36 
11.4067593 
B 
1.22 
45181667 
36 
prime 
B or D or J 
13.28 or 6.19 or 29.0 
45414683 
36 
61.744503 
B 
13.10 
45830431 
36 
31.491.3011 
B 
20.6 
46049041 
36 
prime 
B 
9.31 
46337843 
36 
4327.10709 
D 
22.14 
47635073 
36 
prime 
F 
19.23 
48292669 
36 
151.319819 
E 
15.29 
48339497 
36 
107.451771 
D 
17.15 
The new covering sets are :
code 
modulus 
covering set 
K 
72 
{3, 5, 7, 13, 17, 19, 37, 433} 
L 
48 
{3, 5, 7, 13, 17, 97, 673} 
M 
48 
{3, 5, 13, 17, 97, 241, 257} 
The last of these does not have the prime 7 as a member. Although the value k = 37158601 is a prime, the Keller cycle associated with it contains seven members that are divisible by 7, the lowest of which is k = 1905955429. In fact, another member of the cycle, k = 16352358743, is divisible by 7^{2}. The direct search algorithm used to locate the Sierpinski numbers listed above ignores numbers divisible by 7, but the nature of the Keller iteration ensures that most of members of the Keller cycle are not, and so once one of these is found, any smaller member of the cycle can be retrieved. An example of this will eventually be forthcoming.
The additional overlaps found are :
154337567  shared by cycles 17 (D) and 25 (F)
226521259  shared by cycles 6 (D) and 29 (J)
236281883  shared by cycles 6 (D) and 29 (J)
282777829  shared by cycles 12 (D) and 18 (J)
300943667  shared by cycles 17 (D) and 25 (F)
343302301  shared by cycles 17 (D) and 25 (F)
1300668683  shared by cycles 8 (G) and 23 (K)
4213021013  shared by cycles 8 (G) and 23 (K)
The overlaps shared by sets D and J also occur as generated values for set B. The overlaps involving G and K are also based around the 37109 interchangeability.
Totals for Sierpinski numbers by covering set are :
Total 
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
M 
332 
213 
44 
7 
14 
36 
8 
2 
1 
1 
2 
1 
2 
1 
where the two overlaps that fall into the search range have been assigned to covering set J.
We now complete the search to 10^{8}. As before, we shall ignore the generated values associated with covering set A.
50236847 
36 
5779.8693 
B 
20.27 
50273851 
36 
prime 
F 
4.33 
50407157 
36 
73.690509 
E 
14.6 
50835497 
36 
23.139.15901 
E 
15.16 
51172253 
36 
11.97.199.241 
E 
14.16 
51299477 
36 
prime 
B 
20.23 
51612259 
72 
149.346391 
N 
30.0 
51642601 
36 
prime 
B 
20.30 
51767959 
36 
3109.16651 
B 
13.5 
51889823 
72 
2347.22109 
P 
31.0 
52109063 
36 
17.257.11927 
B 
20.13 
52343539 
36 
131.463.863 
B 
13.21 
53085709 
36 
79.671971 
B 
1.29 
54345857 
36 
prime 
E 
7.28 
55218901 
36 
71.777731 
E 
11.35 
55726831 
36 
41^{2}.33151 
B 
20.34 
55876981 
36 
1373.40697 
E 
7.19 
56191673 
72 
79.711287 
P 
31.48 
56330011 
36 
173.325607 
B 
20.18 
56517767 
36 
7013.8059 
F 
4.30 
56777509 
36 
127.447067 
E 
14.35 
56924089 
36 
113.503753 
E 
11.13 
57396979 
36 
2557.22447 
B 
1.21 
57410477 
36 
127.251.1801 
D 
6.11 
57451021 
36 
389.147689 
E 
15.35 
57616051 
36 
1021.56431 
B 
13.27 
57732559 
36 
83.695573 
B 
13.9 
57798079 
36 
17.3399887 
E 
15.21 
57816799 
36 
53.1090883 
F 
32.0 
57940433 
36 
821.70573 
B 
20.5 
59198569 
72 
prime 
K 
33.0 
59929127 
36 
479.125113 
F 
4.14 
60097043 
36 
5653.10631 
E 
11.6 
60143641 
36 
prime 
B 
20.26 
60303137 
36 
prime 
E 
15.4 
60610801 
36 
prime 
E 
7.7 
60666107 
36 
19.3192953 
F 
32.3 
60909197 
36 
89.684373 
B 
13.4 
61079749 
36 
167.365747 
B 
20.12 
61196987 
36 
prime 
B 
13.20 
62012387 
36 
131.473377 
E 
15.12 
62888633 
36 
prime 
B 
20.33 
63190223 
36 
23.2747401 
B 
20.17 
63662611 
36 
5381.11831 
F 
25.31 
63676073 
36 
prime 
D 
17.13 
63723707 
36 
71.897517 
B 
1.20 
63833243 
36 
4951.12893 
B 
13.26 
63891497 
36 
887.72031 
B 
13.8 
65623711 
36 
prime 
B 
13.19 
66620329 
36 
2267.29387 
B 
20.16 
66887071 
36 
929.71999 
B or F 
1.19 or 32.14 
66941839 
36 
prime 
B 
13.25 
67510217 
36 
103.655439 
D 
17.11 
67800683 
48 
19.3568457 
L 
24.25 
67837073 
36 
2749.24677 
B 
13.18 
68468753 
36 
179.382507 
B or D or J 
1.18 or 17.9 or 34.0 
68496137 
36 
prime 
B 
13.24 
68574271 
36 
prime 
E 
14.21 
68708387 
36 
11.61.102397 
D 
17.7 
69169397 
36 
11.227.27701 
E or D 
11.20 or 22.4 
69265069 
36 
67.1033807 
F 
19.30 
70207549 
36 
2557.27457 
B 
1.1 
70312793 
36 
601.116993 
D 
22.6 
70364663 
36 
prime 
B 
1.2 
70415327 
36 
41.1717447 
E 
14.30 
70678891 
36 
101.699791 
B 
1.3 
71151293 
36 
31.163.14081 
D 
12.31 
71307347 
36 
31.313.7349 
B or D or J 
1.4 or 17.31 or 34.22 
71408993 
36 
17.4200529 
E 
15.26 
71476051 
36 
137.521723 
D 
22.31 
71768941 
36 
151.461.1031 
E 
14.13 
72553787 
36 
587.123601 
E 
15.32 
72564259 
36 
1223.59333 
B 
1.5 
74343527 
36 
5591.13297 
E 
14.10 
74433497 
36 
17.23.190367 
B 
1.32 
74886377 
36 
17.41.107441 
D 
22.8 
75037639 
48 
31.2420569 
C 
5.16 
75070789 
36 
557.134777 
B 
9.1 
75078083 
36 
2699.27817 
B 
1.6 
75202613 
36 
prime 
B 
9.22 
77564731 
36 
71.1092461 
F 
4.17 
77807131 
48 
prime 
L 
24.6 
78240377 
36 
prime 
D 
12.33 
78816559 
36 
prime 
B 
1.33 
78864593 
36 
643.122651 
D 
17.33 
79539409 
36 
523.152083 
D 
22.33 
80091143 
36 
11.59.123407 
B 
9.2 
80105731 
36 
5737.13963 
B 
1.7 
80354791 
36 
11.7304981 
B 
9.23 
81196967 
36 
31.2619257 
E 
15.20 
81722987 
36 
179.456553 
D 
12.25 
82346449 
36 
89.925241 
E 
11.5 
83304121 
36 
1063.78367 
E 
15.11 
83460571 
36 
71.367.3203 
F 
32.6 
84319681 
36 
173.487397 
B 
9.15 
85442453 
36 
241.354533 
B 
1.26 
86420389 
36 
11.7856399 
B 
13.1 
86585063 
36 
17.317.16067 
E 
14.20 
87376127 
36 
127.397.1733 
F 
32.23 
87505591 
36 
31.2822761 
E 
14.29 
87582683 
36 
41.2136163 
B 
1.34 
88574821 
36 
prime 
E 
15.31 
89469691 
36 
17.5262923 
E 
14.9 
90131851 
36 
113.797627 
B 
9.3 
90161027 
36 
11.59.138923 
B 
1.8 
90600893 
36 
prime 
B 
9.6 
90659147 
36 
17.317.16823 
B 
9.24 
90834301 
36 
2437.37273 
B 
1.11 
92452757 
36 
31.173.17239 
B 
13.32 
92732027 
48 
19.37.131909 
C 
5.11 
92896411 
36 
347.267713 
E 
15.19 
93180713 
36 
4793.19441 
D 
22.10 
94353229 
36 
prime 
B 
9.9 
94741307 
36 
5471.17317 
D 
17.27 
94751851 
36 
1109.85439 
D 
22.27 
94819261 
36 
1153.82237 
B 
9.27 
95562473 
36 
103.927791 
B 
20.1 
95590459 
36 
1163.82193 
E 
14.19 
96050723 
36 
7487.12829 
E 
14.28 
96181013 
36 
1151.83563 
B 
13.14 
96220493 
36 
17.53.269.397 
B 
1.14 
97032773 
36 
41.2366653 
E 
14.8 
98588927 
36 
499.197573 
B 
9.16 
98746133 
36 
2011.49103 
E 
15.18 
98915393 
48 
37.1097.2437 
L 
24.21 
99287341 
36 
6451.15391 
D 
12.20 
99694493 
36 
41.2431573 
D 
22.22 
The new covering sets are :
code 
modulus 
covering set 
N 
72 
{3, 5, 7, 17, 19, 37, 109, 241} 
P 
72 
{3, 5, 7, 17, 19, 37, 73, 241} 
Additional overlaps found including new cycles are :
66887071  shared by cycles 1 (B) and 32 (F)
114921271  shared by cycles 7 (E) and 32 (F)
122311103  shared by cycles 22 (D) and 32 (F)
133228283  shared by cycles 7 (E) and 32 (F)
192413177  shared by cycles 22 (D) and 32 (F)
In the B & F overlap, the subsets {5,19} and {5,109} are interchangeable. The overlaps involving sets E and F are as a result of a {3,19} and {3,73} interchange. Since 73 has the smaller exponent, the primary covering set is F.
From the data obtained, it is increasingly obvious that for each different covering set there will be an even number of distinct Keller cycles. This conjecture is definitely true of set A, since the 2P barrier has been reached. Continuing the search to 1.5*10^{8} would pass the 2P barrier for set B. At the moment, there are 4 Keller cycles associated with set B, and it is unlikely that any more will be found. There are also 4 distinct Keller cycles currently known for sets D, E and F, with no more expected. Justification for the conjecture is as follows.
Suppose k is a Sierpinski number with covering set P = {p_{1}, p_{2}, …} and modulus m and such that k < 2P. Let e_{i} be the exponent of p_{i} to base 2, (so that lcm(e_{1}, e_{2}, …) = m) and let the Nash congruences associated with P be (p_{i}, r_{i}). The jth step of the Keller iteration k ® (2k+P) mod 2P changes the Nash congruences to (p_{i}, r_{i}  j), in effect, sliding the divisibility properties one place to the left. Since the r_{i} are all reduced modulo e_{i} , the mth step realigns all the congruences and we arrive back at k. Thus we have the Keller cycle relating m different Sierpinski numbers. However, rather than simply shifting the congruences by 1, we may consider flipping the congruences about the midpoint m/2. In this case, the new congruences are (p_{i}, m+1 r_{i}) associated with a mirrorimage or flip Sierpinski number k'. The second term here can be replaced by e_{i}+1 r_{i} since e_{i} divides m. Repeating this flip returns the original values. We have k.+1 º 0 mod p_{i} and so k º  mod p_{i }. Replacing the exponent, we have k' º  mod p_{i} . We then have a set of simultaneous congruences that can be solved to obtain k' < 2P, k odd, which always exists by the Chinese Remainder Theorem. We can then generate a flip cycle using the Keller iteration.
As an example, k = 271129 has covering set A, modulus 24, and Nash congruences obtained using psieve are (3,1) (5,0) (7,2) (13,6) (17,6) and (241,10) and Keller label 2.0. Since the primes involved have exponents 2, 4, 3, 12, 8 and 24 respectively, the flipped Nash congruences are (3,0) (5,1) (7,2) (13,7) (17,3) and (241,15), leading to the set of congruences k' º 2 mod 3, k' º 2 mod 5, k' º 5 mod 7, k' º 7 mod 13, k' º 2 mod 17, k' º 211 mod 241. The smallest solution of this is k' = 271577, which is the smallest value in the only other Keller cycle for set A (Keller label 3.0). Thus there is a direct match between members of the two Keller cycles, that is, the jth Keller iterations are flips of each other for all j < 24.
A more complicated example is provided by k = 78557 (Keller label 1.0), the smallest Sierpinski number. This has covering set B, modulus 36, and Nash congruences (3,0) (7,1) (5,1) (73,3) (13,11) (19,15) and (37,27), giving flip congruences (3,1) (7,0) (5,0) (73,7) (13,2) (19,4) and (37,10). Evaluating the flip Sierpinski number, we find k' = 29024869, which has Keller label 9.29. In other words, the flip does not directly match up the Keller labels and requires 29 Keller iterations to do so. We shall call the number of Keller iterations between the base values of the two Keller cycles the flip offset. The flip of 9.0 is, unsurprisingly, found to be 1.29. Rather than slide left 29 places, we could slide right 7 places, by replacing Nash congruences (p_{i}, r_{i}) with (p_{i}, r_{I}+7). However, there is no simple iteration that performs a slide to the right.
In the above example, the number of Keller iterations required is odd, and so there is no label position at which flips match. However, for k = 8184977, labelled 13.0, and which also has covering set B, the flip k' = 66620329 has Keller label 20.16. Since the offset is even, there is a position at which matching labels are flips of each other. This happens at the 8^{th} iteration, that is, Sierpinski numbers 13.8 and 20.8 are flips of each other.
The only problem with our conjecture is the possibility that the flipping process might map every congruence onto itself, and hence k º k' (to within Keller cycles). Suppose the flip is selfmapping, and let p be a member of the covering set such that p has largest exponent e_{p}. The Nash congruence (p,r) flips to (p,r+x) for some offset of x. Therefore, we require m+1 r º r x mod e_{p}, that is, 2r º 1 x mod e_{p}. Since there must be other Nash congruences covering all positions from 1 to m, there must be another prime, q say, such that q covers position r+1. By symmetry, the congruence (q,r+1) flips to (q,r+x 1). Hence m+1 (r+1) º r+x 1 mod e_{q}, that is 2r º 1 x mod e_{q} . In general, each prime p_{i} ¹ p in the covering set covers position r+y_{i} for some value y_{i}, and we find that 2r º 1 x mod e_{i} for all i. Now, using the Keller iteration, we can suppose that r = m, since otherwise we can find another member of the Keller cycle for which this is the case. Thus we have x º 1 mod e_{i} for all i. Now, if e_{i} is even, x must be odd, and viceversa, and hence the e_{i} are either all odd or all even. Also, if there is a prime whose exponent is m, we must have x = 1 since by definition we have x < m.
Suppose the e_{i} are all odd. Then the primes 3 and 5 are immediately ruled out of the covering set. In fact, there are comparatively few primes with odd exponent. The first few are 7, 23, 31, 47, 71, 73, 79, 89, 103, 127, 151, 167, 191, 199, 223, 233, 239. The smallest covering sets that do not include 5 are of modulus 72, there being 7 of these, and they all include 3. The smallest covering set that does not include 3 has modulus 180 and has 16 primes, whose product is approximately 2*10^{30}. All covering sets of modulus 180 include either 3 or 5. There are modulus 360 covering sets that contain neither 3 nor 5, but most of the other primes in these sets have even exponent. A covering set made up entirely of primes with odd exponent would have a huge modulus.
Suppose the e_{i }are all even. Then they must all be powers of 2 since otherwise we could replace primes of order e_{i} with primes of odd exponent f_{i} such that f_{i} divides e_{i}. Consider modulus m = 64. There is only one covering set, consisting of all the primes dividing 2^{64}  1, and one selfflipping Keller cycle, whose smallest value is offset by 29 when flipped.
Proceeding with the flip idea, we can pair up all the known Keller cycles, and also find new ones that lie outwith the current search range. For instance, set H (modulus 48) is associated with only one Keller cycle for k < 10^{8}, starting at k = 11206501. The Nash congruences are (3,1) (7,2) (5,2) (17,0) (257,4) (241,12) and (97,28). The flip congruences are (3,0) (7,2) (5,3) (17,1) (257,13) (241,13) and (97,21). Solving the generated simultaneous congruences gives k' = 8407945943. The Keller cycle produced from this has smallest value k'' = 597875869, which is therefore labelled as the zeroth member of the cycle, and k' is 5 Keller iterations from this. Thus we have found a matching Keller cycle for the one we already knew, with an offset of 5.
For completeness, of the 34 Keller cycles found during the search over k < 10^{8}, we have the following flipping pairs, using label numbers and giving a flip offset.
A 
2 & 3 
offset 0 
B 
1 & 9 
offset 29 

13 & 20 
offset 16 
C 
5 & 10 
offset 27 
D 
6 & 22 
offset 20 

12 & 17 
offset 17 
E 
7 & 14 
offset 19 

11 & 15 
offset 33 
F 
4 & 25 
offset 19 

19 & 32 
offset 13 
L 
24 & 28 
offset 7 
The two Keller cycles 8 and 26 associated with set G do not constitute a pair, but are both paired with cycles outwith the current search range. The same is true of the two cycles associated with set K, though the standard calculations lead to a base value k = 115281169, not much beyond the current limits, whose Keller cycle pairs with cycle 23.
Note that using the alternative congruences (p_{i}, m r_{i}) as the flip congruences instead of
(p_{i}, m+1 r_{i}) does not change the pairing of Keller cycles, but increases the offset by 1 (modulo m).
We now continue the search to 2*10^{8}. For covering set B, 2P(B) is reached within this range, and no generated values are listed.
100093157 
36 
prime 
E 
14.18 
100323289 
36 
11. 
E 
14.27 
100387913 
36 
prime 
B 
20.21 
100834471 
36 
prime 
B 
1.27 
102310339 
36 

D 
17.20 
102790343 
36 
prime 
B 
13.2 
102832981 
36 

B 
20.10 
103812287 
36 
41. 
D 
6.23 
104697533 
36 
61. 
F 
19.11 
105114931 
36 
59. 
B 
1.35 
105885947 
72 

R 
35.0 
106330741 
36 
11. 
B 
9.35 
106363697 
36 
11.31. 
B 
9.20 
106596713 
36 
101. 
D 
12.35 
107177209 
36 
prime 
E 
7.1 
107432603 
36 
113.167. 
F 
25.34 
108628343 
72 
151. 
S 
36.0 
108923657 
36 
prime 
B 
1.24 
109093577 
36 
23. 
D 
17.35 
109168141 
36 
107.197. 
B 
13.35 
109758563 
36 

E 
7.2 
110213267 
36 
127. 
B 
9.4 
110271619 
36 
31.139.157.163 
B 
1.9 
110676233 
36 
prime 
B 
13.30 
110825251 
36 
prime 
D 
12.12 
111151351 
36 
prime 
B 
9.7 
111267859 
36 
23. 
B 
9.25 
111608297 
36 
97. 
B 
13.12 
111618167 
36 
prime 
B 
1.12 
111792841 
36 
prime 
D 
22.35 
112787573 
36 
prime 
E 
7.30 
113271289 
36 
17.223. 
B 
20.8 
114145729 
36 
53. 
B 
9.33 
114596513 
36 
337. 
F 
4.28 
114855079 
36 
101. 
B 
13.33 
114921271 
36 

E or F 
7.3 or 32.34 
115281169 
72 

K 
37.0 
115449353 
36 

F 
4.12 
115566797 
72 

R 
35.44 
116138629 
36 

D 
6.32 
116279749 
36 

E 
11.1 
117188839 
36 

F 
32.12 
118656023 
36 

B 
9.10 
118912069 
36 

E 
7.21 
119588087 
36 

B 
9.28 
119863547 
48 

C 
10.7 
119879899 
48 

C 
5.24 
120527153 
36 

D 
12.27 
120979291 
36 

E 
7.31 
121074511 
36 

B 
20.2 
122091961 
48 

C 
10.26 
122311103 
36 

D or F 
22.18 or 32.21 
122311591 
36 

B 
13.15 
122390551 
36 

B 
1.15 
122514181 
36 

E 
14.1 
122685113 
36 

E 
7.14 
123100501 
36 

E 
11.15 
124371917 
36 

B 
9.12 
124463569 
36 

E 
7.25 
125208229 
36 

E 
15.1 
125246687 
36 

E 
7.4 
125773231 
36 

D 
6.18 
126351319 
36 

D 
22.13 
126596461 
36 

E 
15.23 
127127419 
36 

B 
9.17 
127963643 
36 

E 
11.2 
128100173 
36 

B 
9.30 
129197389 
36 

F 
32.32 
129764281 
36 

F 
32.10 
130725391 
36 

B 
20.22 
130896953 
36 

B 
20.29 
131044847 
36 

F 
32.19 
131618507 
36 

B 
1.28 
133228283 
36 

E or F 
7.22 or 32.17 
134045869 
36 

B 
20.4 
135147473 
36 

B 
20.25 
135229937 
36 

F 
25.20 
135530251 
36 

B 
13.3 
135615527 
36 

B 
20.11 
135792317 
36 

E 
11.8 
136519969 
36 

B 
20.32 
136616693 
36 

E 
15.6 
137021401 
36 

B 
13.7 
137362727 
36 

E 
7.32 
137536591 
36 

D 
17.6 
137847349 
36 

E 
7.9 
138385817 
36 

B 
20.15 
138411353 
36 

E 
11.26 
138836071 
36 

D 
17.30 
138920423 
36 

D 
22.30 
138994189 
36 

B 
13.17 
139051463 
36 

F 
25.22 
139310029 
36 

B 
1.17 
139323721 
36 

B 
13.23 
139630819 
36 

F 
19.18 
140432507 
36 

E 
14.2 
140733241 
36 

E 
11.31 
140774371 
36 

E 
7.15 
141605147 
36 

E 
11.16 
143453693 
36 

E 
15.14 
144043891 
36 

D 
12.24 
144331283 
36 

E 
7.26 
144975841 
36 

E 
11.11 
145820603 
36 

E 
15.2 
145897519 
36 

E 
7.5 
146053577 
72 

G 
38.0 
146880319 
48 

C 
5.40 
148597067 
36 

E 
15.24 
150553051 
36 

D 
17.26 
150558323 
36 

D 
22.26 
151060223 
48 

C 
10.13 
151331431 
36 

E 
11.3 
151570849 
36 

E 
15.9 
152106751 
48 

C 
5.30 
152252267 
36 

F 
19.33 
154337567 
36 

D or F 
17.19 or 25.24 
155088541 
36 

D 
6.22 
155223473 
72 

S 
39.0 
155825641 
36 

E 
14.25 
156654991 
36 

F 
19.20 
156701453 
72 

P 
40.0 
157539121 
36 

E 
7.35 
158595023 
36 

D 
12.11 
161416097 
36 

E 
7.12 
161860711 
36 

E 
7.23 
163378771 
48 

C 
5.14 
164337949 
36 

D 
22.17 
165025171 
36 

F 
4.25 
165347657 
36 

E 
11.24 
165928129 
36 

E 
11.29 
166069013 
36 

D 
6.17 
166358057 
36 

D 
22.12 
166988779 
36 

E 
11.9 
167604349 
36 

F 
25.17 
168637531 
36 

E 
15.7 
169073869 
60 

T 
41.0 
169701229 
36 

E 
14.23 
170129599 
36 

E 
7.33 
170337737 
72 

K 
37.27 
171098843 
36 

E 
7.10 
171950693 
36 

D 
17.5 
172081733 
36 

E 
11.22 
172226851 
36 

E 
11.27 
172600433 
36 

D 
17.29 
172642609 
36 

D 
22.29 
174329011 
60 

U 
42.0 
175204343 
36 

D 
12.23 
176269159 
36 

E 
14.3 
176870627 
36 

E 
11.32 
176952887 
36 

E 
7.16 
177065453 
36 

E 
14.32 
178458923 
36 

D 
17.25 
178461559 
36 

D 
22.25 
178614439 
36 

E 
11.17 
180351181 
36 

D 
17.18 
181040117 
36 

E 
15.28 
181339441 
48 

C 
5.38 
182097361 
36 

E 
14.5 
182311531 
36 

E 
15.15 
182384417 
48 

C 
10.11 
182479909 
36 

D or E 
12.10 or 14.15 
182646049 
48 

C 
5.28 
184066711 
36 

E 
7.27 
184503233 
36 

E 
11.34 
184832273 
36 

E 
7.18 
185282537 
36 

E 
14.34 
185355827 
36 

E 
11.12 
185619293 
36 

E 
15.34 
187045351 
36 

E 
15.3 
187199183 
36 

E 
7.6 
190784569 
36 

D 
12.22 
191478481 
36 

E 
11.19 
192411859 
36 

D 
17.24 
192413177 
36 

D or F 
22.24 or 32.27 
192598279 
36 

E 
15.25 
192778253 
36 

E 
14.12 
198067007 
36 

E 
11.4 
198545843 
36 

E 
15.10 
199388327 
36 

D 
17.23 
This range provides us with the first Sierpinski numbers of modulus 60. Covering sets and sample Sierpinski numbers with modulus 60 have been obtained using alternative methods previously, but the smallest such numbers had not.
The new covering sets are :
code 
modulus 
covering set 
R 
72 
{3, 5, 7, 13, 17, 37, 109, 433} 
S 
72 
{3, 5, 7, 13, 17, 19, 73, 433} 
T 
60 
{3, 5, 7, 11, 13, 41, 61, 151, 331} 
U 
60 
{3, 5, 7, 11, 13, 31, 41, 61, 331} 
Additional overlaps found including new cycles are :
8664092933  shared by cycles 23 (K) and 39 (S)
9863014727  shared by cycles 36 (S) and 37 (K)
9936760217  shared by cycles 36 (S) and 37 (K)
11968332053  shared by cycles 8 (G) and 35 (R)
21530959441  shared by cycles 8 (G) and 35 (R)
24733157753  shared by cycles 8 (G) and 35 (R)
The overlaps involving sets K and S result from interchangeability of the subsets {3,5,37} and (3,5,73}. Since 73 has smaller exponent than 37, set S is the primary covering set. The G & R overlap is caused by interchanging the sets {5,19} and {5,37}.
If we continue, the limit 2P(E) = 209191710 is reached quickly and no new cycles with covering set E are found. The following list takes us as far as 2P(D) = 412729590.
200019049 
36 

F 
25.9 
201181193 
36 

E 
15.30 
202876561 
36 

D 
17.22 
205410169 
36 

E 
14.7 
206266849 
36 

E 
15.17 
206940361 
36 

E 
14.17 
207055427 
36 

E 
14.26 
207140783 
36 

F 
19.35 
208234613 
36 

D 
6.1 
208884353 
36 

D 
6.25 
210104431 
36 

D 
6.2 
210465533 
72 

P 
43.0 
210586403 
72 

P 
40.36 
211062227 
72 

V 
44.0 
211073063 
72 

K 
23.21 
211403911 
36 

D 
6.26 
212497043 
72 

W 
45.0 
213447751 
48 

I 
21.28 
213578567 
48 

I 
46.0 
213844067 
36 

D 
6.3 
214767383 
72 

P 
43.48 
215481983 
36 

F 
25.26 
216443027 
36 

D 
6.27 
218039041 
48 

C 
10.30 
218649563 
36 

F 
4.20 
220022057 
36 

D 
12.1 
221323339 
36 

D 
6.4 
224129461 
72 

G 
26.24 
224751679 
36 

F 
19.22 
225632671 
48 

L 
24.14 
226521259 
36 

D or J 
6.28 or 29.9 
228505373 
48 

L 
28.17 
230009513 
36 

D 
17.1 
230667589 
36 

D 
12.6 
232190537 
48 

C 
10.23 
233679319 
36 

D 
12.2 
235566677 
36 

F 
19.29 
236281883 
36 

D or J 
6.5 or 29.22 
236936209 
36 

D 
12.14 
240806569 
36 

D 
22.1 
240936503 
72 

K 
47.0 
245952859 
48 

C 
10.20 
246677723 
36 

D 
6.29 
252919021 
36 

D 
6.14 
253282909 
36 

F 
19.10 
253424869 
48 

C 
10.44 
253654231 
36 

D 
17.2 
254970383 
36 

D 
12.7 
258189721 
36 

D 
6.34 
258232399 
36 

F 
4.27 
258658819 
36 

F 
4.11 
260993843 
36 

D 
12.3 
265532837 
36 

F 
32.31 
265816283 
36 

F 
32.9 
266142407 
60 

X 
48.0 
266198971 
36 

D 
6.6 
267507623 
36 

D 
12.15 
268549111 
36 

F 
25.19 
272308613 
36 

D 
6.9 
275248343 
36 

D 
22.2 
275743817 
36 

D 
12.29 
278770729 
48 

Y 
49.0 
282777829 
36 

D or J 
12.18 or 18.13 
282879617 
36 

D 
22.20 
284736317 
36 

F 
25.16 
286604449 
48 

L 
28.44 
286990651 
36 

D 
6.30 
290215763 
48 

L 
24.17 
291966617 
48 

C 
10.47 
292099127 
60 

X 
50.0 
293678719 
48 

C 
10.32 
296728129 
36 

D 
6.20 
297140731 
36 

F 
32.26 
297988073 
60 

X 
51.0 
299040481 
36 

D 
22.15 
299473247 
36 

D 
6.15 
300943667 
36 

D or F 
17.3 or 25.8 
301946329 
48 

L 
28.40 
303043789 
36 

D 
17.16 
303126409 
72 

S 
36.51 
303575971 
36 

D 
12.8 
308587501 
72 

N 
30.34 
308914459 
72 

Z 
52.0 
310014647 
36 

D 
6.35 
313197379 
48 

L 
28.20 
314416021 
72 

K 
37.6 
315622891 
36 

D 
12.4 
318717481 
36 

F 
19.28 
321185749 
36 

D 
6.12 
326033147 
36 

D 
6.7 
327575597 
36 

F 
19.9 
328650451 
36 

D 
12.16 
332320309 
60 

X 
53.0 
333700561 
36 

F 
32.30 
333716941 
36 

D 
17.14 
338252431 
36 

D 
6.10 
338513269 
48 

L 
28.12 
341371831 
48 

AA 
54.0 
341385229 
36 

D 
17.12 
343302301 
36 

D or F 
17.10 or 25.15 
343401623 
48 

L 
24.41 
343781569 
36 

D 
17.8 
344131891 
36 

D 
22.3 
344703589 
36 

D 
22.5 
345122839 
36 

D 
12.30 
346990381 
36 

D 
22.7 
348667381 
36 

D 
12.32 
348979489 
36 

D 
17.32 
349316897 
36 

D 
22.32 
350284703 
48 

C 
10.25 
351639391 
48 

L 
28.26 
354064201 
72 

Z 
52.38 
356137549 
36 

D 
22.9 
359190863 
36 

D 
12.19 
359292259 
48 

AB 
55.0 
359394439 
36 

D 
22.21 
360292883 
36 

F 
19.27 
361019063 
72 

G 
8.56 
362845549 
36 

D 
12.34 
364093981 
36 

D 
17.34 
364721941 
36 

F 
19.8 
365443613 
36 

D 
22.34 
365928503 
72 

AC 
56.0 
367616507 
36 

D 
6.31 
367784423 
36 

F 
32.29 
369699767 
72 

R 
57.0 
369810769 
36 

D 
12.26 
372585293 
36 

F 
25.14 
379908443 
48 

C 
5.37 
380430931 
48 

C 
10.10 
380561747 
48 

C 
5.27 
383295113 
36 

F 
19.7 
384932701 
48 

L 
24.46 
387091463 
36 

D 
6.21 
387226789 
36 

F 
25.13 
388176109 
72 

P 
31.39 
391716167 
36 

D 
22.16 
392581699 
36 

D or F 
6.16 or 19.6 
392726221 
36 

D 
22.11 
394547537 
36 

F 
25.12 
395522539 
36 

D 
17.4 
395847409 
36 

D 
17.28 
395868497 
36 

D 
22.28 
398207911 
36 

F 
25.11 
398258243 
48 

C 
10.29 
399722783 
36 

D 
17.17 
400787147 
36 

D 
12.9 
402513331 
36 

F 
4.1 
403158377 
36 

F 
4.2 
404448469 
36 

F 
4.3 
404939477 
36 

D 
12.21 
405333991 
48 

C 
10.22 
405753781 
36 

D 
22.23 
407028653 
36 

F 
4.4 
408182963 
72 

G 
26.43 
410985473 
36 

D 
17.21 
412189021 
36 

F 
4.5 
The new covering sets are :
code 
modulus 
covering set 
V 
72 
{3, 5, 7, 17, 37, 109, 241, 433} 
W 
72 
{3, 5, 7, 13, 19, 109, 241, 433} 
X 
60 
{3, 5, 7, 11, 13, 31, 41, 61, 151} 
Y 
48 
{3, 5, 7, 13, 97, 241, 673} 
Z 
72 
{3, 5, 13, 17, 19, 37, 109, 241} 
AA 
48 
{3, 5, 7, 17, 97, 257, 673} 
AB 
48 
{3, 5, 13, 17, 97, 241, 673} 
AC 
72 
{3, 5, 7, 13, 19, 37, 241, 433} 
Note that we have run out of convenient letters to identify covering sets, and it is appropriate to develop a new nomenclature.
Two of the new covering sets do not include the prime 7. In fact, the value of 52.0 was initially overlooked by the algorithm since it divisible by 7, and only located by generating the Keller cycle from 52.38. It is possible that other Sierpinski numbers have been similarly overlooked, but, as mentioned previously, will eventually be discovered. However, it is likely that 52.0 is the smallest Sierpinski number divisible by 7.
Another interesting development is the occurrence of 4 closely grouped values all with covering set X but all in different cycles.
An additional overlap occurs between cycles with covering sets K & R. Many mnore overlaps between members of Keller cycles and generated values exist.
The current allocation of covering sets to Keller cycles, ignoring set J, is as follows:
set 
modulus 
cycles 
status 
A 
24 
2, 3 
(complete) 
B 
36 
1, 9, 13, 20 
(complete) 
C 
48 
5, 10 

D 
36 
6, 12, 17, 22 
(complete) 
E 
36 
7, 11, 14, 15 
(complete) 
F 
36 
4, 19, 25, 32 

G 
72 
8, 26, 38 

H 
48 
16 

I 
48 
21, 46 

K 
72 
23, 33, 37, 47 

L 
48 
24, 28 

M 
48 
27 

N 
72 
30 

P 
72 
31, 40, 43 

R 
72 
35, 57 

S 
72 
36, 39 

T 
60 
41 

U 
60 
42 

V 
72 
44 

W 
72 
45 

X 
60 
48, 50, 51, 53 

Y 
48 
49 

Z 
72 
52 

AA 
48 
54 

AB 
48 
55 

AC 
72 
56 

We now continue to 5*10^{8}.
415951157 
48 

C 
10.43 
422509757 
36 

F 
4.6 
422590909 
72 

P 
31.15 
425780911 
48 

I 
46.35 
426694847 
36 

F 
19.1 
435119569 
48 

L 
24.12 
435222031 
48 

C 
10.46 
435711979 
36 

F 
19.14 
443151229 
36 

F 
4.7 
450231953 
72 

G 
26.51 
451521409 
36 

F 
19.2 
456324301 
48 

I 
46.27 
457592539 
36 

F 
25.1 
460059647 
36 

F 
25.28 
464561807 
48 

L 
24.39 
466621249 
48 

L 
28.24 
469555673 
36 

F 
19.15 
472729967 
36 

F 
4.22 
478078081 
48 

AA 
58.0 
482420221 
72 

G 
59.0 
484434173 
36 

F 
4.8 
488303521 
60 

X 
60.0 
489719779 
36 

F 
25.5 
494851853 
48 

L 
24.37 
497138431 
36 

F 
19.24 
497214301 
48 

C 
10.42 
497369479 
60 

X 
61.0 
There are 3340 distinct Sierpinski numbers identified to this point, of which 657 belong to at least one Keller cycle. This includes only one, namely 308914459, that is divisible by 7.