Cover structures by base

In recent weeks I have been rereading the 2000s file ‘Generalizing Sierpiński Numbers to Base b’, written by a team from the University of Tennessee at Martin.

Although Sierpiński numbers to base 2 are well-known in studies of prime numbers because Proth primes — those of the form k*2ⁿ+1 — occur frequently as possible factors of binomial numbers like Mersenne and Fermat numbers, similar numbers for other bases were not studied until the 1990s. Moreover, although a major project exists to verify the smallest Sierpiński number for all bases up to 1030, bases as small as 71 have not been started yet.

What was really interesting to me re-reading ‘Generalizing Sierpiński Numbers to Base b’ (my off-line .pdf copy is under a slightly different title but has the same text) this summer was the discussion of various covering set periods. The UT Martin team noted that different bases have vastly different minimal periods for a covering set of primes to repeat. For bases that are 2 or one fewer than a power of 2, this period is relatively long, since for any b of the form 2ⁿ-1, b²-1 has no “primitive divisor” — that is, no prime divisor that does not divide a smaller number of the form bⁿ-1. This is because:

1. b²-1 = (b+1)(b-1)
2. b+1 is a power of 2
3. both b+1 and b-1 divide by 2
4. thus, b²-1  lacks a prime divisor for these bases
5. Bang’s Theorem states that the only other such case is 2⁶-1, which I will not discuss further

The absence of a factor with period 2 means that for bases of the form 2ⁿ-1 covering sets must be built from prime numbers with longer periods. For other numbers, there will always be a cover with period 12 or shorter, but for base 3, there can be no cover repeating more frequently than every 48 terms, as was established by Yannick Saouter in 1995.

In the following table, I will indicate the presence or absence for bases from 2 to 175 of covers with the following periods:

• 2
• 3
• 4
• 6
• 8
• 9
• 10
• 12
• 15

Covers with periods 5, 7, 11 and 13, as was noted by the UT Martin team, do not occur for any base so small as 175. 14-covers have not been investigated, although such a cover would be expected to involve eight primes, one with period 2 and seven with periods 7 or 14.

8-, 9-, 10- and 15-covers were not discussed in the UT Martin study, although I have long known of the 8-cover {11, 73, 101, 137} in base 10. The smallest base for which an 8-cover provides the smallest Sierpiński number is 168, while I know of no base where a 9-cover, 10-cover or 15-cover provide the smallest Sierpiński or Riesel number. Nevertheless,  my re-read made me feel these were worthy of study.

For all bases:

• red means the base lacks a cover with that period
• light green means a non-primitive cover
• dark green means a primitive cover that cannot be reduced

Presence or absence of N-cover for Bases from 2 to 175
N-cover	2	3	4	6	8	9	10	12	15
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175

The amazing figures of Jack Owens in 1926 and 1930

Although I have ever since childhood been interested in (Australian rules) football statistics, it is only since reading the Full Points Footy site in the 2000s that my interest was extended beyond the VFL/AFL. Whilst I did have some background knowledge of other football competitions, it was only with Full Points Footy that I gained any detailed knowledge of the SANFL and WA(N)FL before the national completion began in the late 1980s.

One of the most remarkable things I discovered from reading the SANFL season scores of the middle to late 1920s was the extent to which Jack Owens monopolised scoring for Glenelg in some seasons. In his book Goals, Goals, Goals: A Study of League and Association Full-Forwards, Marc Fiddian paid limited attention to cases of players kicking all their team’s goals in a match, in favour of focusing on the extremely large goal tallies common in the VFA during my childhood. Nevertheless, there is an attraction to cases where one player monopolises scoring to an unusual degree — the team is totally dependent upon one player to score, and that naturally affects the attention granted to that player, and may make that player work much harder knowing that nobody else can do his job. (The phenomenon is well known to me from such county bowlers as “Tich” Freeman and George Dennett, who often took over half their team’s wickets for an entire season, feasible for a spin bowler under conditions not rigged against them by covering and “log bats”.)

Owens, who came from Broken Hill to Glenelg at a time when the club had not won a game in three seasons, would established himself very quickly around the time the club finally opened its account in the first game of 1925. He would set SA(N)FL goalkicking records — that would almost immediately be broken by North Adelaide’s Ken Farmer — in 1927 and 1928. Although Glenelg would improve substantially from their first three seasons, this improvement would during the period under discussion not be sufficient to lift them above seventh in an eight-club competition. What has attracted my attention for some years, though, is the proportion of Glenelg’s goals Owens kicked. The 1989 3AW Book of Footy Records, which I discarded many years ago, noted Ted Tyson’s goalkicking feats for the West Perth team of 1938 and 1939, who lost 27 consecutive games. Whilst these are remarkable, my study of the scores and season goalkicking figures for the SA(N)FL seasons between 1925 and 1932 shows that Owens actually surpassed Tyson’s monopolisation of the scoring. In fact, in the seasons of 1926 and 1930, Owens kicked over half of all Glenelg’s goals, a feat which the 2000s AFL Record encyclopedias show never to have been achieved over a full season in the VFL/AFL. The closest approach occurred in 1944 when Melbourne’s Fred Fanning kicked 48.33 percent of the team’s goals in the fourteen games he played. Fanning, though, missed four through suspension.

1926:

Round	1	2	3	4	5	6	7	8	9	10	11	12	13	14	Total
Owens	0	1	5	4	6	6	7	1	5	7	7	3	6	6	64
	0.00%	20.00%	55.56%	44.44%	75.00%	50.00%	70.00%	14.29%	71.43%	63.64%	43.75%	75.00%	66.67%	66.67%	54.24%
C. Sallis			1	1		3	1	3	1	2					12
Drew			1					1		2	2	1		1	8
Hack			1	1		1	1								4
P. Leverington		1						1			2				4
C. Hoft											1		2	1	4
Lyne	1	1				1									3
W. Hill				2	1										3
Gun						1		1	1						3
Exley		1					1							1	3
L. Sallis			1								1				2
L. Leverington											2				2
J. Davies											1		1		2
Toms	1														1
Lloyd		1													1
C. Hill				1											1
Barbary					1										1
Total	2	5	9	9	8	12	10	7	7	11	16	4	9	9	118

1930:

Round	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	Total
Owens	5	6	4	10	1	5	8	5	5	1	5	3	4	10	8	11	8	99
	71.43%	60.00%	44.44%	90.91%	11.11%	41.67%	53.33%	55.56%	71.43%	16.67%	35.71%	33.33%	40.00%	83.33%	80.00%	68.75%	80.00%	56.25%
Thompson						1	2		1	3	3				1	1		12
Link		2			1		2			2			3					10
Sallis								1			2	3		1		1		8
Rosewarne		1			2	2										2	1	8
Coller			3		2	2												7
Winkler				1	2						1	1	1					6
Morrow					1	1		1										3
Hill			1				1	1										3
Drew							2		1									3
Griffiths		1									1		1					3
K. Oliver												1		1			1	3
Johnston								1				1						2
Whitehead											1		1					2
Guiney			1													1		2
Edwards	1																	1
Percy	1																	1
Wickham						1												1
Lloyd											1							1
Wright															1			1
Total	7	10	9	11	9	12	15	9	7	6	14	9	10	12	10	16	10	176

From these two tables, we can see that:

1. in the last four games of 1930, Owens kicked 37 of Glenelg’s 48 goals —more than 75 percent, a proportion unrivalled for so long a period
2. in 1926, Owens kicked at least half of Glenelg’s goals in nine of fourteen matches (64.28 percent)
3. in 1930, Owens kicked at least half of Glenelg’s goals in ten of seventeen matches (58.82 percent)

With Tyson, John Coleman, Austin Robertson junior, Peter Hudson, Tony Lockett, and Brendan Fevola, scoring monopolisation was never as pronounced as 1), although, apart from “Ocker”, I have not thoroughly checked. It is odd that 3AW did not notice this.

Why no anarcho-capitalism before the 1970s?

In Commerce and Coalitions, Ron Rogowski argued on pages 165 to 168 that socialism never evolved in the United States because of the enduring abundance of land and scarcity of labour. Whilst Rogowski is too timid about discussing race — for a start, the theories of Eli Heckscher and Bertil Ohlin, and later Wolfgang Stolper and Paul Samuelson, are in no way inconsistent with division on the basis of race if ownership of production factors be divided thereby — he does offer quite reasonable explanations in terms of how labour loses from trade and becomes either timid or powerless except during falling trade.

One related point I thought of on reading Rogowski but which has not been discussed is why anarcho-capitalism evolved where and when it did. Since the Republican Revolution, anarcho-capitalism has been a highly influential ideology amongst the US ruling class, with Randolph Hohle arguing that anarcho-capitalist and allied writers use a code of:

“private” = “White”

“public” = “Black”

to justify their vision of a society without public services. (Although Hohle says this code developed only after the civil rights revolution of the middle 1960s, I feel that it probably dates back to at least the early Republic when free blacks, as I noted previously, became viewed as wholly deviant.)

Despite its influence, anarcho-capitalism has never been subject to detailed scholarly study. The critical question is why, given the benefits not having to pay taxes would have to the extremely rich, anarcho-capitalism only became an established ideology in the 1970s, and never evolved in Europe or Japan at all. Here, Rogowski can provide a very clear answer. Anarcho-capitalism is, by definition, a militant movement of the upper classes to remove the power of labour, and in its goal to eliminate the public sector is the direct opposite of socialist movements aiming to eliminate the private sector. By inverting what Commerce and Coalitions says about socialism, we can predict logically that anarcho-capitalism (or similar militant movements against democracy) will arise where(ever) capital benefits from expanding trade and labour does not, that is, where capital is abundant and labour scarce. In Rogowski’s terminology, anarcho-capitalism will arise in advanced economies with a high land:labour ratio. Under these conditions, capital and land gain from free trade, whilst labour sees its opportunities restricted and its incomes reduced. Thus class struggle is intense as capital and land seek to eliminate the power of groups opposed to their unlimited power by funding politicians so dedicated, and ultimately to destroy all possible sources of power of such groups. Under this condition, those who do not lie at the very bottom of society — lacking the resources to unite with those who do as their skills are devalued — will stigmatise those at the very bottom. Rogowski himself showed that this stigmatisation can be extremely powerful on page 85 of his 1974 Rational Legitimacy: A Theory of Political Support, and on page 157 he suggests that the probability of a coalition between lower classes with different stigmatisation states is likely to be very low if the groups lack combined economic power.

As I have discussed here, there were never any advanced economies with even a relatively high land:labour ratio at any point between the birth of Christ and about 1900. An undated file from the University of Michigan noted this at PS 489.1 (page 5):

“Abundance of Land and Capital, Scarcity of Labor (p. 32):

Who?

1. 1840: no one
2. 1914: US and Canada
3. [1991-: US and Canada, Australia and New Zealand, Arab Gulf States, Russia, Kazakhstan, Turkmenistan, Turkey (marginal)]”

Thus, before 1900 no state favourable to the development of strong and/or influential anarcho-capitalist type movements existed. Although Proudhon’s ideas as depicted by Marxist Left Review strikingly resemble anarcho-capitalism in many respects, if Rogowski be correct then Proudhon was planting an ideological seed onto the unfavourable soil of mid-nineteenth-century France.

In the developed countries of northwestern Europe, enduring abundance of labour meant that when trade expanded, enriched workers pressed for and gained more rights and a larger public sector. Additionally, labour did not oppose capital because both benefitted from trade, and this made capital compromise severely with labour’s opposing demands for the highest possible taxes and greatest regulation. This is even more emphatically true of later-developing East Asia.

In a labour-rich society, labour opposes capital only when capital is scarce. There capital becomes militant only during falling trade and is intensely protectionist. This militant capital will then support autarky with only the most restricted possible trade, a position entirely opposed to free-trading anarcho-capitalism. Unlike anarcho-capitalism, Marxian socialism was and is not explicitly pro-free trade. Thus one could theoretically imagine militant socialism arising in advanced, land-rich economies under falling trade, though history does not provide an example and there exist many reasons why this would not happen even if the falling trade were extremely prolonged. Thus, “no anarcho-capitalism in labour-rich societies” is almost certainly a more rigid rule than “no militant socialism in labour-scarce societies”.

The rule that there is no anarcho-capitalism in labour-abundant societies can also be extended to other forms of ruling class militancy like religious fundamentalism. This also first emerged in the United States after it became abundant in capital, notably with the second Ku Klux Klan, and re-emerged there after the postwar globalisation. It has been the dominant force in the Gulf monarchies since the pivotal oil crises of the 1970s. These movements argue, at least implicitly, that God’s law decrees a natural hierarchy based on non-class distinctions like race, gender or religion, and that this is a natural order for all human societies.

The problem of “don’t ask” just from fearing no

This evening I went into a terrible temper tantrum when my mother complained that I left cooking rissoles for tonight’s dinner until they were seriously overcooked and smelling.

The tantrum, with familiar phrases (mantras) like:

“get the word “can” out of your head”

“there’s no such word as “can””

and less familiar mantras like

“I used to be able to [remember to come down] but I can’t now”

and gestures like trying to place the word “can” in the rubbish bin in the kitchen [by pointing to the rubbish bin in a terribly agitated manner], became so bad that I even walked out of the house without my mobile phone! Outside I was just as angry and agitated — trying basically to make my mother recognise that I was totally incapable of reminding myself that I had cooking to remember when I was sitting at the very computer where I am writing this post.

My mother was just as upset as I was — and I might note that she has not been well in recent weeks, having mucus-y coughs for the past week or more. It has been difficult for me to appreciate her troubles, but I hope I becoming less bad at it.

After I came back — deliberately knocking softly in order to, as it were, persuade my mother that I had calmed down — I did speak much more calmly about the issue. I enjoy cooking, but my mother had said that I might have to give it up if I forget it time and time again as I have been doing for the past few weeks.

When I look at it logically, I recognise how I was obsessed with watching football and possessed zero desire to do the cooking. I simply wished to watch the football and listen to music as I had been doing for many hours before I went out to walk the dog, but feared my mother’s opinion too much to say “no”. So, I did the cooking as quickly as possible and was quite lucky in fact not to burn the spring onion, celery and mushroom mix that was to be mixed with the mince to make the rissoles. Although I said to Mummy that I would come back when I put the rissoles in the oven, I immediately and totally forgot as soon as I was back settled at the computer watching football. There was not the slightest thought about what was happening inside the oven until the DVD was finished. So unconcerned was I that I was actually surprised that the rissoles had nearly burned over!

Once I came back, realising that I could not keep screaming and screaming outside, and even feeling my vocal cords somewhat weakened, I tried to talk to my mother about what needs to be done to give some hope this will not happen as consistently as it has been of late. I am probably reasonably convinced that there is no hope I can be instructed to remember to come down when I am totally focused on listening to music or on watching football (of course, anything I am watching on the computer would have the same effect as football!). Both of us were extremely upset at the way I expected my mother to be a “lackey” upon whom we can fall back whenever I forget cooking jobs that I do just to please her when my actual desire is to listen to music or watch something on the computer. I actually say I want to cook to appease my mother when I actually want to spend all day doing exactly what I want — which, today, was to watch football.

However, the most notable lesson from tonight’s tempera tantrum was that my consistent refusal to ask my mother whether she would either:

1. remind me by coming up to the study or
2. look after my cooking

out of fear of being told “no” is really, really bad. It might be wise because I know all too well that if my mother said “no” I would most likely go straight into a meltdown as bad as today’s. However, my deliberate silence has meant that my mother and I are both left ignoring the cooking with risks of burning. When I look at it, I thought of the many old English cricketers — for example Walter Brearley or George Louden — who would likely have greatly improved England’s prospects in Australia but were never questioned about touring because MCC was sufficiently aware they had zero possibility of being away from business for so long. My case is different because I have never tried — out of partially justifiable fear of being told “no” — to ask my mother to watch over my cooking. Instead, I assume she will do so as my slave. When in a tantrum, of course, I justify it as a “one-way street” where I can do and get exactly what I want one hundred percent of the time, although logically this “one-way street” is totally selfish and makes everybody else totally immaterial. It is clear to me — whether I cam capable of change or not (and I have zero belief I am) — that I must do something to prevent me forgetting my cooking, or develop some sort of compromise where I do the preparation and my mother agrees to watch over the cooking. In a logical mood these compromises are terribly unfair to my mother, but given my lack of genuine interest in cooking it is unclear what else we can do??