Searching for large Sierpinski numbers
In this article's immediate predecessor, I described the results of a survey of Sierpinski numbers, which consisted mainly of enumerating these to a limit of 5*10^{8} and highlighting any issues that were raised along the way. In the current article, rather than build up data gradually, I will take a more global view of the results and matters arising from a survey to the extended limit of 2*10^{9}.
The search method is as before, where each possible value of k, not divisible by 3, 5 or 7, is tested for each exponent n from 1 to 1000. If any prime is found for a particular k, that value of k is ignored. All values of k which survive this step are then passed to the program psieve3.exe, which builds up a list of Nash congruences. If at any time, the accumulation of Nash congruences combine to produce a complete covering, then we have a Sierpinski number, and the associated covering set can be extracted from the congruences (though not without the occasional difficulty). Covering sets are labelled, generally according to the order they were first identified.
The complete list of covering sets associated with at least one Sierpinski number under 2*10^{9} is as follows (where P is the product of the primes in the set):
code 
modulus 
covering set 
2P 
A 
24 
{3, 5, 7, 13, 17, 241} 
1.1e7 
B 
36 
{3, 5, 7, 13, 19, 37, 73} 
1.4e8 
C 
48 
{3, 5, 7, 13, 17, 97, 257} 
1.1e9 
D 
36 
{3, 5, 7, 13, 19, 73, 109} 
4.1e8 
E 
36 
{3, 5, 7, 13, 19, 37, 109} 
2.1e8 
F 
36 
{3, 5, 7, 13, 37, 73, 109} 
8.0e8 
G 
72 
{3, 5, 7, 13, 17, 19, 109, 433} 
4.2e10 
H 
48 
{3, 5, 7, 17, 97, 241, 257} 
2.2e10 
I 
48 
{3, 5, 7, 13, 17, 257, 673} 
8.0e9 
K 
72 
{3, 5, 7, 13, 17, 19, 37, 433} 
1.4e10 
L 
48 
{3, 5, 7, 13, 17, 97, 673} 
3.0e9 
M 
48 
{3, 5, 13, 17, 97, 241, 257} 
4.0e10 
N 
72 
{3, 5, 7, 17, 19, 37, 109, 241} 
6.6e10 
P 
72 
{3, 5, 7, 17, 19, 37, 73, 241} 
4.4e10 
R 
72 
{3, 5, 7, 13, 17, 37, 109, 433} 
8.1e10 
S 
72 
{3, 5, 7, 13, 17, 19, 73, 433} 
2.8e10 
T 
60 
{3, 5, 7, 11, 13, 41, 61, 151, 331} 
3.8e12 
U 
60 
{3, 5, 7, 11, 13, 31, 41, 61, 331} 
7.7e11 
V 
72 
{3, 5, 7, 17, 37, 109, 241, 433} 
1.5e12 
W 
72 
{3, 5, 7, 13, 19, 109, 241, 433} 
5.9e11 
X 
60 
{3, 5, 7, 11, 13, 31, 41, 61, 151} 
3.5e11 
Y 
48 
{3, 5, 7, 13, 97, 241, 673} 
4.3e10 
Z 
72 
{3, 5, 13, 17, 19, 37, 109, 241} 
1.2e11 
AA 
48 
{3, 5, 7, 17, 97, 257, 673} 
6.0e10 
AB 
48 
{3, 5, 13, 17, 97, 241, 673} 
1.0e11 
AC 
72 
{3, 5, 7, 13, 19, 37, 241, 433} 
2.0e11 
AD 
72 
{3, 5, 7, 13, 37, 73, 241, 433} 
7.7e11 
AE 
144 
{3, 5, 7, 13, 17, 19, 73, 97, 577} 
3.6e12 
AF 
72 
{3, 5, 7, 17, 19, 73, 109, 241} 
1.3e11 
AG 
72 
{3, 5, 7, 13, 17, 37, 73, 433} 
5.4e10 
AH 
144 
{3, 5, 7, 13, 17, 19, 73, 257, 577} 
9.5e12 
AI 
72 
{3, 5, 7, 13, 17, 73, 109, 433} 
1.6e11 
AJ 
48 
{3, 5, 7, 13, 97, 241, 257} 
1.6e10 
AK 
72 
{3, 5, 7, 17, 37, 73, 109, 241} 
2.5e11 
AL 
72 
{3, 5, 7, 13, 19, 73, 241, 433} 
4.0e11 
AM 
72 
{3, 5, 7, 17, 19, 37, 241, 433} 
2.6e11 
AN 
60 
{3, 5, 7, 11, 13, 31, 41, 61, 1321} 
3.1e12 
AP 
48 
{3, 5, 7, 17, 241, 257, 673} 
1.5e11 
AR 
60 
{3, 5, 7, 13, 31, 41, 61, 151, 331} 
1.1e13 
AS 
72 
{3, 5, 13, 17, 19, 73, 109, 241} 
2.4e11 
AT 
180 
{3, 5, 7, 11, 13, 19, 37, 61, 151, 181, 631} 
2.2e16 
AU 
72 
{3, 5, 13, 17, 19, 37, 73, 241} 
8.2e10 
AV 
48 
{3, 5, 7, 17, 97, 241, 673} 
5.6e10 
AW 
60 
{3, 5, 7, 11, 13, 31, 61, 151, 1321} 
1.1e13 
AX 
72 
{3, 5, 7, 17, 19, 73, 241, 433} 
5.2e11 
AY 
144 
{3, 5, 7, 17, 37, 73, 97, 109, 257} 
2.6e13 
AZ 
72 
{3, 5, 13, 17, 37, 73, 109, 241} 
4.7e11 
BA 
48 
{3, 5, 13, 17, 97, 257, 673} 
1.1e11 
BB 
180 
{3, 5, 7, 11, 13, 19, 31, 41, 73, 181, 631} 
6.0e15 
BC 
60 
{3, 5, 7, 11, 13, 31, 41, 151, 331} 
1.9e12 
The letter J is missing from this list for historical reasons, as it was once used to explain the concept of overlaps (Sierpinski numbers belonging to more than one Keller cycle).
The above list includes 12 of the 15 available covering sets of modulus 48, but surprisingly only 7 of the 23 available covering sets of modulus 60. No Sierpinski numbers of modulus 96 occur within the current limit.
The Keller cycle produced by the Keller iteration k ® (2k+P) mod 2P (where P is the product of the primes in the covering set) is a closed loop of length equal to the modulus of the covering set.
For a given covering set, once the 2P limit is reached, there can be no more Keller cycles and so the number of these is finite. Additionally, the concept of the flip cycle has already been introduced. Amongst other things, this guarantees that for any covering set whose constituent primes have order modulo 2 that are not all even or not all odd, there must be an even number of Keller cycles. To the current search limit of 10^{9}, this has been verified to be true for covering sets A, B, C, D, E and F.
The current allocation of covering sets to Keller cycles is as follows:
set 
modulus 
# cycles 
status 
A 
24 
2 
(complete) 
B 
36 
4 
(complete) 
C 
48 
2 
(complete) 
D 
36 
4 
(complete) 
E 
36 
4 
(complete) 
F 
36 
4 
(complete) 
G 
72 
4 

H 
48 
2 

I 
48 
2 

K 
72 
4 

L 
48 
2 

M 
48 
2 

N 
72 
3 

P 
72 
4 

R 
72 
3 

S 
72 
4 

T 
60 
2 

U 
60 
2 

V 
72 
1 

W 
72 
1 

X 
60 
12 

Y 
48 
2 

Z 
72 
2 

AA 
48 
2 

AB 
48 
1 

AC 
72 
2 

AD 
72 
1 

AE 
144 
1 

AF 
72 
3 

AG 
72 
4 

AH 
144 
1 

AI 
72 
1 

AJ 
48 
2 

AK 
72 
2 

AL 
72 
2 

AM 
72 
2 

AN 
60 
4 

AP 
48 
1 

AR 
60 
1 

AS 
72 
1 

AT 
180 
1 

AU 
72 
1 

AV 
48 
1 

AW 
60 
2 

AX 
72 
1 

AY 
144 
1 

AZ 
72 
1 

BA 
48 
1 

BB 
180 
1 

BC 
60 
1 



114 

I have no explanation as yet for the high incidence of cycles with covering set X.
There are 13394 distinct Sierpinski numbers in the range considered, including 141 overlaps.
This includes only four values, namely 308914459, 1319709979, 1554424697 and 1905955429 that are divisible by 7.
The smallest Sierpinski numbers for each of the moduli identified so far are:
modulus 
smallest Sierpinski number 
24 
271129 
36 
78557 
48 
327739 
60 
169073869 
72 
1777613 
144 
516108143 
180 
1143502909 
One value, namely 987152869, is the base value for two distinct Keller cycles, both of modulus 60. This is the only occurrence of such in the range considered.
My previous numbering of Keller cycles was based on the order in which they were found. However, the cases of the Sierpinski number divisible by 7, and the increasing number of overlaps, sometimes means that in order to retain a rising sequence, some reordering has to take place every now and then. To avoid this, I have simply listed the base values in increasing order.
The full list of Keller cycle base values, with associated covering sets, is as follows:
sequence 
base number 
modulus 
covering set 
factorisation 
1 
78557 
36 
B 
17*4621 
2 
271129 
24 
A 
prime 
3 
271577 
24 
A 
59*4603 
4 
322523 
36 
F 
prime 
5 
327739 
48 
C 
prime 
6 
934909 
36 
D 
prime 
7 
1290677 
36 
E 
137*9421 
8 
1777613 
72 
G 
29*61297 
9 
2510177 
36 
B 
167*15031 
10 
5455789 
48 
C 
59*89*1039 
11 
5841947 
36 
E 
271*21557 
12 
6828631 
36 
D 
23*337*881 
13 
8184977 
36 
B 
prime 
14 
8959163 
36 
E 
prime 
15 
10306187 
36 
E 
prime 
16 
11206501 
48 
H 
prime 
17 
11822359 
36 
D 
prime 
18 
12413281 
36 
F 
17*191*3823 
19 
12756019 
36 
B 
prime 
20 
15273751 
48 
I 
prime 
21 
17220887 
36 
D 
3709*4643 
22 
22024609 
72 
K 
1303*16903 
23 
24885199 
48 
L 
113*191*1153 
24 
27862127 
36 
F 
29*960763 
25 
36029731 
72 
G 
71*507461 
26 
37158601 
48 
M 
prime 
27 
41134369 
48 
L 
prime 
28 
51612259 
72 
N 
149*346391 
29 
51889823 
72 
P 
2347*22109 
30 
56191673 
72 
AU 
79*711287 
31 
57816799 
36 
F 
53*1090883 
32 
59198569 
72 
K 
prime 
33 
105885947 
72 
R 
3181*33287 
34 
108628343 
72 
S 
151*719393 
35 
115281169 
72 
K 
311*370679 
36 
146053577 
72 
G 
4721*30937 
37 
155223473 
72 
S 
37*4195229 
38 
156701453 
72 
P 
3931*39863 
39 
169073869 
60 
T 
53*163*19571 
40 
174329011 
60 
U 
59*257*11497 
41 
210465533 
72 
P 
181*1162793 
42 
211062227 
72 
V 
prime 
43 
212497043 
72 
W 
11*19317913 
44 
213578567 
48 
I 
127*1681721 
45 
240936503 
72 
K 
1129*213407 
46 
266142407 
60 
X 
23*347*33347 
47 
278770729 
48 
Y 
2551*109279 
48 
292099127 
60 
X 
37*2659*2969 
49 
297988073 
60 
X 
prime 
50 
308914459 
72 
Z 
7*41*101*10657 
51 
332320309 
60 
X 
29^{2}*73*5413 
52 
341371831 
48 
AA 
2711*125921 
53 
359292259 
48 
AB 
131*2742689 
54 
365928503 
72 
AC 
347*1054549 
55 
369699767 
72 
R 
43*8597669 
56 
389737669 
48 
AP 
prime 
57 
478078081 
48 
AA 
13*41^{2}*131*167 
58 
482420221 
72 
G 
71*6794651 
59 
488303521 
60 
X 
prime 
60 
497369479 
60 
X 
19*683*38327 
61 
510877279 
72 
AD 
11*79*587891 
62 
516108143 
144 
AE 
prime 
63 
563461069 
72 
AF 
97*1051*5527 
64 
577256123 
72 
AG 
43*13424561 
65 
597875869 
48 
H 
19*23*1031*1327 
66 
605248493 
144 
AH 
2351*257443 
67 
613997417 
48 
AJ 
11*673*82939 
68 
615884447 
72 
P 
1523*404389 
69 
620629981 
72 
AI 
3461*179321 
70 
627253343 
72 
N 
prime 
71 
689546903 
72 
AK 
277*2489339 
72 
693557633 
72 
AL 
211*3287003 
73 
710530043 
60 
X 
7507*94649 
74 
757599167 
72 
N 
13*58276859 
75 
764612281 
48 
AJ 
17*557*80749 
76 
769883761 
72 
S 
23*5449*6143 
77 
832047023 
72 
R 
17047*48809 
78 
839316707 
60 
X 
17*1109*44519 
79 
844627607 
72 
AM 
1901*444307 
80 
879092101 
48 
Y 
1109*792689 
81 
914079949 
60 
AN 
19*3491*13781 
82 
974552641 
60 
AN 
151*379*17029 
83 
987152869 
60 
U 
337*2929397 
84 
987152869 
60 
AR 
337*2929397 
85 
1017439067 
48 
M 
41*24815587 
86 
1021957141 
72 
AS 
prime 
87 
1038777953 
72 
S 
prime 
88 
1051180381 
72 
AG 
prime 
89 
1077331111 
60 
AN 
prime 
90 
1077974903 
60 
AN 
6637*162419 
91 
1088472887 
72 
AZ 
59*199*92707 
92 
1143502909 
180 
AT 
17*239*431*653 
93 
1156418821 
60 
X 
59*19600319 
94 
1161643607 
72 
AG 
439*2646113 
95 
1181603461 
72 
AG 
107*929*11887 
96 
1269436171 
48 
AV 
23*55192877 
97 
1316626321 
60 
T 
73*18035977 
98 
1329539273 
72 
AM 
35461*37493 
99 
1366677863 
72 
Z 
20269*76427 
100 
1373855461 
72 
AC 
11*43*2904557 
101 
1452979757 
60 
AW 
prime 
102 
1460644561 
60 
X 
prime 
103 
1471655687 
72 
AL 
79*193*263*367 
104 
1502556901 
72 
AX 
101*14876801 
105 
1550990453 
72 
AF 
61*4967*5119 
106 
1591804639 
60 
X 
17*37*73*34667 
107 
1642059527 
72 
AK 
prime 
108 
1644987931 
72 
AF 
499*3296569 
109 
1695587059 
144 
AY 
prime 
110 
1828217513 
60 
X 
677*2700469 
111 
1840951877 
48 
BA 
2467*746231 
112 
1850694047 
180 
BB 
prime 
113 
1945610987 
60 
BC 
prime 
114 
1946251501 
60 
AW 
prime 
Sometimes because of an overlap, a nonbase value is identifed before a base value for certain Keller cycles. This explains the occasional inconsistency in the labelling of covering sets (e.g. AJ before AI, the positioning of AP, etc.).
I have created a comprehensive list of every Sierpinski number to the limit of 10^{9}. In this list, individual Sierpinski numbers are labelled according to their position in the associated Keller cycle, with base (or smallest) values given an index of 0. Generated values, that is, Sierpinski numbers obtained by using the open iteration k ® (k+2P) are not individually labelled, but simply identified by the code of their covering set.