“pet room Archie’s”!

Today, attempting to comment upon a recent post by Robert Spencer in Jihad Watch — one, in fact, where I was pleased that he was showing a small inclination to discuss something utterly pivotal to fighting jihad — I saw that I was having my writing corrected, amazingly I felt than and now, to

“pet room Archie’s”!

I laughed so hard at “pet room Archie’s” — a correction repeated twice — that I had to tell my mother and brother, and I still feel amazed.

The correct word I intended to type, of course, was “petromonarchies”. As the graph below from Google Ngram shows, the word “petromonarchies” has existed for almost half a century to describe the Persian Gulf states of Baḥrain, Kuwait, Qaṭar, Oman, Saudi Arabia and the United Arab Emirates, alongside the Bornean state of Brunei.

Use over time of the word “petromonarchies” and the phrases “oil monarchy” and “oil monarchies” according to Google Ngram. The singular “petromonarchy” was not found

The word “petromonarchies” has its first documented use (from Ms. magazine) in 1982. That ought to be enough for the web to recognise it, even if the graph above does show “petromonarchies” to have become a more frequently used word since the middle 2000s. “Pet Room Archie’s” sounds so artificial one wonders how it came about unless these were seen as the only possible words with the group of consonants p-t-r-m-r-ch-s.

What is amazing is not merely that “pet room Archie’s” is so funny, but how frivolous it seems. What a “pet room Archie” would be is not clear. What petromonarchies are is anything but frivolous. They are a massive problem for the planet and for the immense majority of the world’s population, because their ability to maintain huge revenue with zero taxes on capital:

1. produces a continuous downward pressure on capital taxation at a global scale
2. prevents the smallest possibility of reducing emissions to curtail climate change
1. this is because doing so would necessarily weaken the power of the petromonarchies
2. as implied by 1), weakening the petromonarchies would reduce or eliminate the downward pressure on capital taxation that has so enriched the world’s richest since the energy crisis
3. allows them to sponsor terrorism that has the effect of weakening the resistance of the world’s lower classes to the upward transfer of wealth made possible by their high revenues and zero taxes on capital

nth year when the desert city finally appears?

Today, despite recent forecasts that a negative IOD would lead to good rainfall over eastern Australia, Melbourne, at a mere 5.4 millimetres, is experiencing its driest September on record after another prediciton of rain overnight “high chance of showers” failed yet again to materialise. In fact, on present trends Melbourne is headed for its driest spring month on record, beating out October 1914 by 1.5 millimetres and November 1895 by 0.6 millimetres. Most likely, Melbourne, despite th forecast of Weatherzone, will see less rain still in October and November. Weatherzone, even said “rain” today but already the sky is clearing!

Despite this Weatherzone forces, no rain has fallen in Melbourne. This will almost certainly be Melbourne’s not merely driest September, but driest spring month and driest year ever — but much, much worse is to come

A fortnight ago, a prediction of 10 millimetres — possibly in fact 15 —of rainfall and heavy thunderstorms was followed up by just 0.6 millimetres of rainfall!

Rainfall for the first sixteen days of September 2025. Note the very dry conditions over the inland southeast, and wet conditions over a central band of NSW and southwestern Australia. Courtesy Australian Bureau of Meteorology

Despite the fact that quite good rainfall over the rapidly drying southwest of Australia has weakened my tendency towards violent temper outbursts over climate change and the three-decades overdue call for expropriation of the fossil fuel polluters to the final cent and immediate transfer of every last cent of their accumulated wealth to renewable energy and compensation for the costs of climate change, the contrast between BOM predictions and actual rainfall in Melbourne — and to a lesser extent the rest of Victoria and the settled parts of South Australia — is so great that something must be utterly wrong with their forecast modelling.

Total Victorian rainfall for the first sixteen days of September, 2025

What has been virtually continuous for most of August and the first half of September is a sustained block that has produced extreme rainfall over Sydney — 530 millimetres over 45 days — and good rainfall over the aridifying southwest of Western Australia. This is, it must be noted, utterly opposite to what Weatherzone is predicting in its analysis of the negative IOD. However, given that based on the late-2000s documentation of Diane J. Seidel, Fu Qiang, William J. Randel and Thomas J. Reichler it is known that the tropics have shifted at least six degrees poleward since 1964 — equivalent to Melbourne shifting to the latitude of Bourke or Coober Pedy— it is to be expected that higher-latitude anticyclonic systems would severely dry out southern Australia.

Given that with a normally wet negative IOD Victoria is experiencing record drought, it can be expected that when the IOD turns positive it will be far, far worse — if that be possible. Two years ago, Melbourne saw just 0.2 millimetres in the last three weeks of September. As we are seeing something equally dry under conditions predicted to be wet, it is virtually certain that — as I have been predicting for two decades and counting — Melbourne’s annual rainfall is headed for an extremely steep decline to levels comparable to historical totals in Australia’s driest deserts or probably even less. So far this year, Melbourne is 7 millimetres below its 1967 record low of 332 millimetres. It is virtually certain that Melbourne will not receive 7 millimetres for the rest of 2025, and will definitively exceedingly rarely — like one year out of twenty or thirty — receive 332 millimetres in a year again. Most years, Melbourne’s annual rainfall will be a tiny fraction of its former 1967 record low, and about what was received in northern South Australia back in 2019.

The extremely low annual rainfall for northern South Australia in 2019 is what will be received in most years from 2026 onwards. Since the late 2000s, it has been known that the tropical belt has shifted sufficiently to place Melbourne in the same zone as the driest areas on this map, and runaway global warming will from now on place Melbourne there without doubt

For the world’s rich capitalists, unfortunately, the hot desert is the best possible environment because it gives bosses the smallest possible subsistence wage to pay workers, as noted two decades ago by Branko Milanovic, Peter H. Lindert and Jeffrey G. Williamson. Desert states can also typically out-earn states in less ecologically fragile regions, and thus avoid having to rely upon taxes, creating downward pressure on capital taxation at a global scale.

Without mass international worker struggle to globally

1. expropriate the super-polluters
1. socialists have long demonstrated that the only just rate of capital taxation is 100 percent's and the only just profit rate zero,
2. justice demands that workers keep every single cent of what they produce as wages 
2. trailing the corporate polluters under “workers’ courts”
1. these would undoubtedly find them guilty of theft for their entire accumulated wealth and profits
2. the corporate polluters would then be locked up or executed 
3. the entire loot [profit] corporate polluters have stolen from workers, other poor people and the environment as profits would be returned to the people and completely redirected to achieving zero emissions in the absolute minimum possible time
3. completely abolishing profit and private ownership

nothing can be done to limit damage from climate change, let alone to limit its extent. This has long been documented by countless journals and magazines from Red Flag to Socialism Today to Liberation News to Organization Theory. 

Eight new sequences

Two years ago, I discussed the phenomenon of possible numbers of primes ending in a particular digit [1, 3, 7, or 9] between 100k and 100k+99

For a long time afterwards, I wanted to actually make centuries with the maximum number of primes ending in each of 1, 3, 7 or 9 into a reasonable sequence. However, my inability to use the software used by professional mathematicians to find centuries with unusual prime configurations, alongside the fact that the software is not normally used to find this sort of configuration, made if difficult, Recently, however, over several days and nights I have managed to list all centuries with seven primes ending in the same digit up to 10¹⁰:

Centuries [100k to 100k+99] Containing Seven Primes Ending in the Same Digit Up to 10¹⁰:

1	80562, 812412, 830407, 1922407, 3221175, 4246561, 4699195, 5163822, 5972635, 6889392, 10824559, 11647131, 12871998, 14414719, 15209863, 17067460, 23006325, 24014386, 30768175, 31608072, 32133534, 37443006, 44082616, 44823097, 44980450, 47731935, 48607213, 54022315, 54958684, 58407306, 65250324, 65672949, 72608932, 76150752, 76171500, 76967232, 82229494, 91365756, 93799572, 94346032, 94632495, 97150540, 98820502, 99906057,...
3	0, 190783, 768052, 1089709, 1844258, 2076875, 2386669, 3003353, 3440953, 3645619, 4992263, 6363542, 7768946, 11771275, 12201733, 14280710, 19513177, 19916480, 21137401, 21365041, 23104154, 25313755, 26820232, 29518856, 31861282, 32615117, 36697244, 43681403, 44142899, 44160098, 45218771, 45965531, 47014733, 48467767, 48732661, 49482781, 50885369, 52788433, 53482777, 58303430, 60432874, 67088696, 70714409, 79505765, 79644008, 83922821, 85742996, 87239311, 90260663, 92520958, 98023460,...
7	224241, 1599235, 3884413, 6898356, 8428813, 8759248, 10924537, 11838333, 12435657, 15219837, 16784694, 19087533, 19821189, 20164873, 21484788, 25402399, 25564267, 28697595, 28701604, 33482947, 34976343, 37674297, 39663438, 41820411, 46977862, 50299747, 61913167, 74271037, 77368852, 79458772, 84251478, 86998843, 93082627, 96209697,...
9	211696, 974015, 1173662, 1225261, 1239646, 1790287, 2159824, 2815148, 2972188, 4179688, 5416987, 6741980, 7300811, 9143479, 9945887, 10982447, 12604004, 15818663, 16486988, 16835336, 19558576, 20439547, 24812921, 25024748, 25910927, 27292792, 33721163, 34760854, 36064070, 36880424, 42930020, 43607125, 43869583, 44703220, 53590126, 54638992, 55794473, 55944560, 58510697, 58944284, 61995880, 62609960, 66563149, 70544810, 71355032, 71403374, 78722777, 83384546, 83826323, 90718103, 94922326, 98440832,...

In addition, owing to the irregularity of the first centuries containing six primes ending in each of 1, 3, 7, or 9, I have decided to tabulate these up to sixteen million — slightly more than the first million primes. Searching for centuries containing six primes ending in 1, 3, 7 or 9 using the PARI software would undoubtedly be possible, but working out the formula to do so would be far beyond what I managed to grasp of PARI seeking centuries with seven primes ending in one digit. Thus, I simply used a table of the first million prime numbers, filtered them by modulus (10), and looked for cases where the difference between six successive primes with a given modulus 10 was less than 100. I then filtered these to identify cases where the last six consecutive primes with the same modulus (10) was 7x, 8x or 9x mod (100). I did have to weed out the 80563^rd century with seven primes ending in 1, but a thorough check by this inefficient method, as well as mere memory of that century, can do this:

Centuries [100k to 100k+99] Containing Six Primes Ending in the Same Digit In First Million Primes:

1	42, 2986, 4437, 6747, 9780, 11134, 12067, 12268, 15462, 23970, 24352, 24597, 24679, 24865, 32913, 36714, 37108, 39070, 39087, 50664, 51900, 54151, 54646, 59869, 61486, 61986, 64428, 67922, 70279, 70585, 84277, 85257, 87633, 90775, 96048, 96646, 104044, 106856, 108156, 117270, 117795, 119046, 133342, 143152, 146023, 146610, 150891, 150929, 151393, 161685,...
3	2054, 2413, 3587, 5362, 6418, 8543, 13583, 16067, 17510, 22298, 24829, 27086, 29174, 31238, 31637, 32815, 36557, 44101, 44205, 50690, 55856, 57307, 59132, 73752, 77639, 79441, 82757, 97577, 103202, 109975, 113480, 115622, 125819, 126931, 128113, 132637, 133682, 134963, 135920, 137590, 139583, 143606, 150925, 154184, 158697, 160835, 162620,...
7	0, 1, 3, 9, 16, 69, 313, 633, 1095, 1209, 2197, 2817, 3655, 4002, 4833, 7813, 10488, 12414, 13485, 13966, 14661, 15535, 19134, 19231, 20947, 21148, 25142, 27049, 28414, 30745, 32702, 36606, 46791, 48291, 49437, 49733, 52967, 55350, 56235, 59492, 75370, 75771, 78186, 79635, 79818, 79885, 80410, 84912, 88363, 88404, 89467, 89818, 91023, 91798, 96166, 98953, 123678, 130377, 131415, 132847, 134137, 136383,...
9	4, 14, 175, 320, 397, 1232, 6043, 6373, 6400, 7211, 11837, 12082, 24183, 28337, 28852, 34178, 44419, 54530, 56156, 58384, 64225, 68056, 71338, 73663, 79742, 82910, 85174, 92523, 95314, 95331, 98525, 101641, 102301, 108040, 117880, 129901, 130852, 147371, 150586, 153361,...

Are prime-poor small centuries “expectedly” so?

On Mersenne Forum, there was a recent response to my previous blog post about moduli (3003), showing that not all centuries that are exceptionally poor in prime numbers have unusually few “potential” primes [“potential” primes being defined as numbers not divisible by 2, 3, 5, 7, 11, or 13 according to modulus (3003), with 3003 being the product of the first four primes coprime to 100].

In order to see whether centuries with abnormally few primes actually have few “potential” primes, I have aimed to test the modulus (3003) of every century up to ten million that has a record-low or equal-record-low number of primes (compared to smaller centuries only of course). These centuries, with their modulus (3003) and “potential” primes, are tabulated below

k	Total primes	mod (3003)	Total possible primes	...1	...3	...7	...9
2	16	2	17	5	5	3	4
3	16	3	19	4	5	6	4
5	14	5	18	4	4	5	5
7	14	7	19	4	5	4	6
9	14	9	18	6	3	6	3
11	12	11	17	4	5	3	5
13	11	13	18	4	5	5	4
21	10	21	15	4	5	3	3
24	10	24	18	4	3	6	5
31	10	31	20	6	4	5	5
41	9	41	17	4	4	5	4
43	9	43	21	4	5	6	6
48	8	48	19	7	4	5	3
59	7	59	19	3	6	5	5
95	7	95	18	4	5	5	4
142	7	142	20	4	5	5	6
165	7	165	18	5	3	5	5
167	7	167	19	5	3	5	6
186	6	186	17	5	3	5	4
188	5	188	17	3	4	5	5
273	5	273	20	6	5	5	4
314	4	314	17	3	5	4	5
356	4	356	19	5	4	6	4
524	4	524	18	4	4	6	4
588	3	588	19	5	5	4	5
695	3	695	17	5	4	4	4
797	3	797	16	4	3	5	4
1430	3	1430	17	5	5	3	4
1559	1	1559	17	4	5	4	4
2683	1	2683	20	5	4	4	7
4133	1	1130	17	5	6	3	3
10048	1	1039	20	5	6	5	4
11400	1	2391	18	5	3	5	5
12727	1	715	21	5	5	7	4
12800	1	788	19	3	6	5	5
13572	1	1560	21	6	4	6	5
14223	1	2211	19	6	4	4	5
14443	1	2431	19	5	3	6	5
14514	1	2502	16	3	4	5	4
14680	1	2668	21	5	5	6	5
14913	1	2901	19	4	4	6	5
15536	1	521	19	4	7	4	4
15619	1	604	20	5	4	4	7
16538	1	1523	17	4	6	3	4
16557	1	1542	18	5	4	5	4
16718	0	1703	20	6	6	3	5
26378	0	2354	17	5	5	3	4
31173	0	1143	17	4	4	5	4
39336	0	297	18	3	4	6	5
46406	0	1361	18	5	5	4	4
46524	0	1479	17	4	3	6	4
51782	0	731	17	4	6	3	4
55187	0	1133	16	3	4	4	5
58374	0	1317	20	5	6	6	3
58452	0	1395	19	6	4	6	3
60129	0	69	20	6	5	6	3
60850	0	790	19	5	5	4	5
63338	0	275	16	4	5	2	5
63762	0	699	19	5	4	6	4
67898	0	1832	17	3	5	4	5
69587	0	518	20	5	5	3	7
71299	0	2230	20	6	4	4	6
75652	0	577	20	5	5	5	5
78035	0	2960	20	4	5	5	6
78269	0	191	19	5	5	4	5
80277	0	2199	18	5	4	5	4
83674	0	2593	19	5	4	5	5
84213	0	129	19	3	5	6	5
89052	0	1965	19	6	4	4	5
95490	0	2397	17	5	4	5	3
97080	0	984	18	5	3	5	5

Results:

If we consider all centuries up to ten million that have an (equal) record low number of primes vis-à-vis preceding centuries, we find that:

“Potential primes”	All centuries	%	“Record few” centuries	%	Difference in %	Difference in ratio
15	4	0.13%	1	1.41%	1.28%	1057.39%
16	46	1.53%	4	5.63%	4.10%	367.79%
17	244	8.13%	17	23.94%	15.82%	294.68%
18	580	19.31%	13	18.31%	-1.00%	94.80%
19	954	31.77%	19	26.76%	-5.01%	84.24%
20	725	24.14%	13	18.31%	-5.83%	75.84%
21	336	11.19%	4	5.63%	-5.56%	50.35%
22	88	2.93%	0	0.00%	-2.93%	0.00%
23	26	0.87%	0	0.00%	-0.87%	0.00%

The table does seem to indicate systematic differences between centuries with the fewest primes up to that point, and all centuries, in terms of the number of potential primes. Whilst it is true that the proportion of moduli (3003) yielding 22 or 23 “potential primes” is very small, it is revealing that no century up to ten million with a record low number of primes has so many. With 19, 20 and 21 “potential primes”, the difference appears to be just as systematic inasmuch as the proportion of centuries with a record small number of primes having so many is consistently fewer than of all centuries. Contrariwise, for 15, 16 and 17 “potential primes”, the proportion is consistently higher for centuries with record few primes than for all centuries. This does imply that centuries with a record small number of primes do have a distinct tendency to have fewer “potential primes” than other centuries, even if the tendency is not consistent enough to apply to every such century.

Three strange fifteen-prime patterns

Looking through prime patterns today, I have come across two very interesting patterns for nine-digit fifteen-prime centuries, and one of six six-digit fifteen-prime centuries that has extraordinarily few possibilities to have so many:

“Full 9” 1,790,288^th Century:

179028701 = 71 × 277 × 9103

179028703 = 7 × 7 × 3653647

179028707 = 11 × 13 × 409 × 3061

1. 179028709 is prime [1.]

2. 179028713 is prime

3. 179028719 is prime [2.]

179028721 = 3373 × 53077

179028727 = 1699 × 105373

179028731 = 7 × 1163 × 21991

179028733 = 13 × 1451 × 9491

179028737 = 43 × 4163459

4. 179028739 is prime [3.]

5. 179028743 is prime

6. 179028749 is prime [4.]

179028751 = 11 × 17 × 31 × 89 × 347

179028757 = 23 × 67 × 116177

7. 179028761 is prime

8. 179028763 is prime

9. 179028767 is prime

10. 179028769 is prime [5.]

179028773 = 7 × 11 × 19 × 79 × 1549

11. 179028779 is prime [6.]

179028781 = 47 × 3809123

179028787 = 7 × 25575541 

12. 179028791 is prime

13. 179028793 is prime

14. 179028797 is prime

15. 179028799 is prime [7.]

Here we see a fifteen-prime century of the form 3k+1 with seven primes ending in 9 (a “full 9” century). The only smaller century with seven primes ending in 9 I know of is from 21,169,600 to 21,169,699, and it is not impossible that the 1,790,288^th century really is the second with seven primes ending in 9, though I have not asked about this yet.

What makes this pattern even stranger is that eight of the fifteen primes — six of the eight that do not end in 9 — are part of a pair of successive prime quadruples. The other two primes, both of which end in 3, also fit a remarkable pattern. In fact, except for 179,028,773, which has several small prime divisors and could be detected from the century’s modulus (231), there are only four decadal patterns for the whole century, each of which occurs multiple times. The only comparable pattern of which I am aware occurs in the fairly well-known 6,328^th century [632713, 632717, 632743, 632747, 632773, 632777]

The 2,527,250^th Century — 15 Primes, None Ending in 7:

252724901 = 11 × 13 × 1767307

1. 252724903 is prime [1.]

252724907 = 17 × 59 × 251969

252724909 = 19 × 19 × 179 × 3911

252724913 = 7 × 5197 × 6947

2. 252724919 is prime [1.]

3. 252724921 is prime [1.]

252724927 = 7 × 13 × 31 × 101 × 887

4. 252724931 is prime [2.]

5. 252724933 is prime [2.]

252724937 = 29 × 281 × 31013

6. 252724939 is prime [2.]

252724943 = 23 × 41 × 283 × 947

7. 252724949 is prime [3.]

8. 252724951 is prime [3.]

252724957 = 83 × 3044879

9. 252724961 is prime [4.]

10. 252724963 is prime [3.]

252724967 = 11 × 22974997

252724969 = 7 × 47 × 768161

11. 252724973 is prime [4.]

252724979 = 13 × 19440383

12. 252724981 is prime [5.]

252724987 = 1669 × 151423

13. 252724991 is prime [6.]

14. 252724993 is prime [5.]

252724997 = 7 × 7 × 71 × 72643

15. 252724999 is prime [4.]

What we see here is a fifteen-prime century with:

1. six primes ending in 1
2. five primes ending in 3
3. no prime ending in 7
4. four primes ending in 9

Although by deletion from several very small centuries with sixteen primes — the fourth and eleventh — one can easily draw up patterns with fourteen primes but no prime ending in one digit, this century with fifteen primes but not one ending in 7 surprised me so much I felt I had to tabulate it. What is strange here is that seven of thirteen composite numbers not divisible by 2, 3, or 5 end in 7.

The 1,573^rd Century — Almost All Potential Primes Prime

This century is much smaller than the previous two, and it is only today that I realised how improbable it was when one tests its numbers modulo not merely 3, but also 7, 11 and 13. Unlike the two larger centuries, I will tabulate all numbers not divisible by 2 or 5 to illustrate:

157201 = 11 × 31 × 461

157203 = 3 × 3 × 17467

1. 157207 is prime

157209 = 3 × 13 × 29 × 139

2. 157211 is prime

157213 = 7 × 37 × 607

3. 157217 is prime

4. 157219 is prime

157221 = 3 × 3 × 3 × 3 × 3 × 647

157223 = 11 × 14293

157227 = 3 × 7 × 7487

5. 157229 is prime

6. 157231 is prime

157233 = 3 × 17 × 3083

157237 = 97 × 1621

157239 = 3 × 3 × 17471

157241 = 7 × 7 × 3209

7. 157243 is prime

8. 157247 is prime

157249 = 67 × 2347

157251 = 3 × 23 × 43 × 53

9. 157253 is prime

157257 = 3 × 3 × 101 × 173

10. 157259 is prime

157261 = 13 × 12097

157263 = 3 × 19 × 31 × 89

157267 = 11 × 17 × 29 × 29

157269 = 3 × 7 × 7489

11. 157271 is prime

12. 157273 is prime

13. 157277 is prime

14. 157279 is prime

157281 = 3 × 103 × 509

157283 = 7 × 22469

157287 = 3 × 13 × 37 × 109

157289 = 11 × 79 × 181

15. 157291 is prime

157293 = 3 × 3 × 17477

157297 = 7 × 23 × 977

157299 = 3 × 52433

Here, we see that all but seventeen numbers, based on divisibility criteria, cannot be prime. Yet, of these seventeen, a remarkable fifteen actually are prime! Looking through tables modulo 21, 231 and finally 3003, I have never seen a case like this century. Despite the fact that nine numbers out of the 26 not divisible by 2, 3 or 5 can be ruled out modulo 7, 11 and/or 13, there are only two other composite numbers in the whole century. Although with centuries above the “eight-digit gap” I have seen this, in those cases all the composite numbers have highly complex factorisations, whereas 157,237 and 157,249 are semiprimes of a type familiar when learning early factorisations.