Friday, 27 October 2023

The “Australia shock”

Following on from his Commerce and Coalitions, of which I wrote several discussions earlier in 2023, I have recently been looking at the preview of Ron Rogowski’s new book Shocking Contrasts: Political Responses to Exogenous Supply Shocks.

In many ways Shocking Contrasts is intended as an expansion of what Rogowski discussed over three decades earlier in Commerce and Coalitions. His earlier book looked at how endowment affect politics in differently-endowed societies, whilst Shocking Contrasts deals with changes in national and global endowments of important production factors. The book covers chiefly the Black Death in late medieval Europe and the blockade and quest for Lebensraum in interwar Germany, but also looks at the post-Máo Zédōng “China shock”, the fifteenth-century invention of the printing press, and the “railway revolution” of nineteenth-century Russia.

Some years ago, my mother liked to say that “the world is made in China”, upon seeing

“Made in China — Fabrique en Chine [French]”

on virtually every product we bought from stores like Myer and David Jones.

If China has a unique abundance of labour, Australia has an even more unique abundance of land. On page 162 of Handbook on East Asian Economic Integration (edited by Kimura Fukunari, Mari Pangestu, Shandre M. Thangavelu and Christopher Findlay), it is noted that World Bank statistics reveal Australia to possess:

  1. twenty-nine times the global average of agricultural land per capita
  2. sixteen times the global average of mineral resource endowments per capita
  3. eighteen times the global average of land per worker

Page 94 of Commerce and Coalitions revealed Australia as having the second-lowest population per unit area of agricultural land, behind only Botswana.

Thus, the opening up of Australia’s land — preindustrially useless because of extreme deficiency of essential nutrients in its ancient soils and absence of the advanced technology required to smelt the lithophile metals (aluminum, manganese, titanium) in which Australia is so rich — would be expected to constitute a shock of increased land supply. This process began in the 1850s as new land was opened for grazing and continued intensely until at least 1975. In many ways, especially in Queensland where new land is still being cleared for farming, this “Australia shock” (as I will call it) continues to the present.

Shocking Contrasts illustrates no example of a supply shock involving gain of land. The “Australia shock” is probably too prolonged to be sufficiently unexpected for inclusion. Gain of land via trade between 1850 and World War I has been noted by Jeffrey Gale Williamson and subsequently Kevin Hjortshøj O‘Rourke as radically reducing rents in land-poor trade-open European nations. However, both those authors focused on the “New World” as a whole, although this “New World” is comprises two radically different groups of lands:

  1. the New World proper, or the Western Hemisphere of the Americas and New Zealand, which are fertile regions sparsely populated because:
    1. they were peopled very late as either ice sheets or deep seas had to be crossed, and
    2. outside the altiplano native large animals were invariably too egalitarian in social structure for domestication
  2. Australia and Africa, more accurately named the “Ancient World”: early-peopled lands whose low fertility and ecological fragility inhibited technological or human cultural development

All the scholars noted previously demonstrate dramatic increases in inequality in the Ancient and New Worlds, and comparably dramatic decreases in the Old World. The situation before World War I is then contrasted with the interwar situation whereby the Ancient and New Worlds experienced egalitarian trends, and the Old World inegalitarian. Nevertheless, even admitting the importance of interwar deglobalisation, technological developments were continuously increasing Australia’s relative abundance of land: for instance, titanium metal could not be smelted commercially until the 1940s, and large areas of southern Australia could not be farmed until chalcophile micronutrient deficiencies were discovered in the 1950s. By then little potential farmland in the temperate New World remained uncleared as remaining land was invariably too wet, cold or steep for any sort of farming, while the hemisphere’s chalcophile ores were rapidly depleting. These trends have accelerated since the middle 1970s.

Evidence:

If there has been an “Australia shock”, we should see evidence that returns to land have demonstrably fallen. Rogowski shows re the “China shock” that wages have fallen and inequality risen even in other labour-abundant nations, suggesting that their labour is out-competed by China.

Falling Returns to Land:

Evidence of falling returns to land ever since the opening of Australia is very well documented for land-scarce Europe and East Asia. However, the trend can also be seen in the economic history of the New World proper, where areas least able to diversify beyond specialisation in agriculture have seen the sharpest declines in global economic rank, as noted by John Singleton for New Zealand, and much more widely for Argentina. Excluding probably Kazakhstan, no nation rivals New Zealand and the Southern Cone for high agricultural land value per capita (see page 162 of Handbook on East Asian Economic Integration and NationMaster’s comparison here for justification).

As Jeffrey G. Williamson noted in the 1990s, increases in total land value in the most land-rich regions under globalisation can reach orders of magnitude. Strong evidence of falling returns to land can be seen in the political strengthening of landless peasants in land-rich parts of Latin America under postwar rising trade, most notably in Nicaragua during the 1970s and Bolivia during the 1990s. This contrasts sharply with the weakness of labour in these regions during the pre-World War I globalisation, and on page 115 of Commerce and Coalitions Rogowski says:

“Nicaragua, as a thinly populated state, should have experienced a radicalization of land rather than labo[u]r”

Despite the growth of trade after World War II, landowners in Latin America were faced with decreasing returns due to to increased global abundance of land and their inability to own newly usable land and ores. Peasants in land-rich nations, who used but did not own land, would benefit from this trend and increasing returns to the labour they owned. Similar trends can be detected in other land-rich Latin American nations, most of which democratised late in the twentieth century, and also in Afghanistan, where the war between the PDPA and the mujahiddin can be understood as an economic conflict between:

  • diminishing returns to land (via the “Australia shock”) and
  • economic gains of abundant land against scarce capital and labour (via expanding trade)
Certain other land-rich nations — notably Algeria and 1970s Ethiopia — experienced similar effects as  users of land over owners became favoured by increasing global abundance of land at a time when climatic conditions in Australia were unusually favourable.

Impoverishment of Backward, Land-Abundant Societies:

More consistent than unpredicted radicalisation of landless peasants in some land-rich societies has been the plain impoverishment of sub-Saharan Africa and Latin America since World War II. Previously, their per capita incomes and living standards were generally higher than labour-rich tropical Asia and even mainland East Asia. However, with the increasing supply of farmland and lithophile minerals from Australia, those regions became faced with a loss–loss scenario. Their comparative advantages in agriculture and non-lithophile minerals declined, whilst their comparative disadvantages in labour-, capital- or skill-intensive industries did not. The result, as has been noted by many economists, is an impoverishment that (as noted above) extended to what before the Second World War were some of the wealthiest nations in the world.

Rogowski in Commerce and Coalitions attributes some of this impoverishment to protectionist policies favouring the urban elite, but gives little idea as to why the free-trading rural sectors failed to gain political control, and zero idea of why they were challenged after they did in many Latin American nations (e.g. by Shining Path in Peru). Sharica Sudan under Omar Bashir and Iran under the ayatollahs, although imperfect cases due to US sanctions (which I previously argued to have only a minor effect), do not suggest “traditionalist” regimes would have fared better than those based upon the capital–labour coalition. If we assume a continuous gain of land from Australia, we do see that countries less land-surfeited than Australia, but unable to specialise elsewhere, will ipso facto be outcompeted and become poorer. Policies designed to increase competitiveness are likely to be difficult to implement. For instance, radical currency devaluations to make exports more competitive with more land-abundant rivals will be painful even for the very richest, and have never been observed even with IMF structural adjustment, as noted by the IMF’s own Independent Evaluation Office in 2007. Similarly, there are limits to which already low farming and mining wages can fall, and to how much these will restore preceding profitability.

Global Warming:

A third aspect of the “Australia Shock”, more obvious than the previous two, is global warming. Australia, is, of course — excluding the Gulf States — the worst offender in greenhouse gas emissions, with per-citizen emissions ten times the global average. However, Australia’s role in global warming is even greater than this because:
  1. it has been documented that Australian flora is the most efficient in the world at storing carbon
  2. increasingly intense forest fires as observed in 2019 and last September are likely to severely effect carbon storage be removing even small areas of fire-sensitive rainforests
  3. Australia has the largest total area of agricultural land in the world and thus the largest area of native vegetation cleared for farming or grazing
  4. the superabundance of land in Australia has led to protein production increasingly using Australian land instead of Asian or European labour
  5. the extreme abundance of lithophile metals whose smelting is extremely energy-intensive has caused them to substantially supplant scarce easily smelted chalcophile metals, and likely eliminated incentives for recycling chalcophiles

The global temperature graph shows a continuous long-term cooling from peak Holocene warmth in the ninth millennium BC until World War I, followed by slow warming during the long mid-century (between the Bolshevik Revolution and 1973 oil crisis) and rapid accelerating warming since the middle 1970s. This does not agree with the peaks in land clearing in Australia — which occurred in the last half of the nineteenth century and the postwar era. However, it is certainly possible that a side-effect of land-clearing in Australia may have been a shift towards land-intensive protein production in the New World, as observed after World War II, for the following reasons:

  • as noted in Richard Seymour’s The Disenchanted Earth: Reflections on Ecosocialism and Barbarism, the US government subsidised beef production massively.
  • it is certainly likely that this demand for animal sources of protein would have been less if labour-intensive plant proteins in the tropics were not rendered so uncompetitive by Australia’s superabundance of grazing land
  • the same increase in beef production took place even in land-poor Europe, based on US government policy with the Marshall Plan
It is also true that the supplanting of chalcophile elements with Australian lithophiles — largely smelted via coal — coincided with global warming accelerating. This also happened immediately after fear of exhaustion of chalcophile “base metals” became acute in the early 1970s.

Thus, even if the raw temperature figures do not imply immediate direct causation, the role of the “Australia shock” in global warming is almost certainly significant.

Secularisation:

Another effect I suspect from the development of Australia’s pre-industrially unusable land is the disappearance of the influence of Christianity, Hinduism and Buddhism on societies previously dominated thereby. Secularisation is a logical effect of any large gain of land, because land is the only factor of production religious institutions can own. Hence, if previously scarce land becomes abundant, religious organisations’ returns will fall dramatically.

The evidence here is highly supportive. The major opening of Australia in the late nineteenth century coincides well with the growth of anti-religion sentiment within the working classes of Europe. The second major “opening” of Australia in the postwar era coincides very well with the complete disappearance of Christianity as a force in Europe, and of Hinduism and Buddhism in land-poor monsoon Asia (where Stalinism and similar ideologies came to dominate). As I noted earlier, even the land-rich New World was by no means exempt from this trend: it simply could never be so land-rich as Australia was.

Also, perhaps more controversially, I can argue that the unpredictability and relative aseasonality of the Australian climate might make religions substantially based upon the very regular (although it must be acknowledged that in places like Gujarat rainy season rainfall is as variable as rainfall in Australia) seasonal cycles of the Levant and monsoon Asia less relevant. Outside the southwestern quarter Australia’s climate is largely controlled by the El Niño Southern Oscillation, which is a highly irregular nonannual cycle with periods as long as seven years and large fluctuations in strength. The climate of the non-ENSO southwestern quarter is vastly less well understood but appears to be most sensitive to acyclical changes in high latitudes like polar ozone depletion and Antarctic pressure and sea surface temperatures. Under such conditions, religious systems developed in Eurasia lose their critical connections to the land and nature upon which their cosmology was based. Because farming in Australia required and requires highly advanced technology to combat ancient soils and a frequently erratic climate, there could never be a replacement religion of a traditional type. Although Australian farmers do speak of attachment to the land, it is extremely important to recognise how unnatural this is because productive crops and livestock require the chalcophile elements so deficient in Australia.

“Australia shock” or European subsidies?

One question I have asked with my putative “Australia shock” is how much its effects are due not to the expansion of land clearing in Australia, but rather to European farm subsidies. Historians such as Jeffrey G. Williamson note that before World War I, although protection of manufacturing was greater than today, agricultural protection was much lower. As noted throughout this post, European and East Asian workers were empowered by nineteenth-century globalisation, and because land was the pillar of ruling class power, subsidising farming became a tool to preserve the influence of farmers against manufacturing workers desiring radical social revolution.

European farm subsidies would logically be expected to have similar global effects to development in Australia. In an artificial manner, they increase the global land supply, and would be expected to reduce returns to land elsewhere in the world, and impoverish backward, land-abundant societies specialising in agriculture. The question is whether European subsidies would have more effect than opening of new land in Australia.

I would argue that this is a debatable question. The critical issue is disincentives to use land efficiently. These are often argued to be a direct result of farm subsidies. Nevertheless, as Wadham Wilson Wood noted in his 1957 Land Utilization in Australia, such disincentives apply equally to superabundant land under a free market, because such land is too cheap and the increase in rent would have to reach extraordinary levels to counter the problem. Politics may be one reason why rent increases have been very limited in rural Australia: Martin Taylor’s Bludgers in Grass Castles from the late 1990s notes how the political influences of pastoralists has made rents “token” in his words. However, limits to productivity on most Australian farmland are severe, with maximum yields for no crop so much as one-half the global maximum, and for some they are as low as one-seventh the global maximum due to the extreme antiquity of Australian soils. This would mean that land values in Australia would never rise nearly so high as in the New World, even under free trade without farm subsidies. Moreover, even if European (and East Asian) farm subsidies do not encourage globally efficient use of land, without the continual opening of new land in Australia up to the twenty-first century the previously noted disincentives would be absent. Hence, the “Australia shock” has itself produced the lavish farm subsidies we see today even in the New World: we cannot entangle it from modern farm subsidies.

Conclusion:

The previously argued points provide substantial evidence for global effects via continued clearing and development of Australia’s superabundant flat land for agriculture. Calling it a “shock” in the sense of Rogowski’s Shocking Contrasts is somewhat inaccurate because the gain of land from opening Australia has occurred over too prolonged a period, but that does not deny its effects.

It is also true that I am uncertain about other effects I expected, chiefly reduced inequality in backward, labour-abundant economies, which I could not relate to the opening of Australia. However, many other predicted effects seem on at least a superficial study to be relatable thereto.

Tuesday, 5 September 2023

Where there cannot be the maximum number of primes

In a few posts from last year, I discussed pentatrigesimal representation of numbers using all numbers and all letters except O (“O” being a common reading of the number “0” and very similar in shape).

One interesting discovery I made is that whilst 35 consecutive numbers can contain as many as ten primes, if we tabulate divisibility of numbers between each multiple of 35, we find something quite interesting:

k 6n 6n+1 6n+2 6n+3 6n+4 6n+5
1
2
2 3 2
2 2
2
2 3
3 3 2
2
2
4 2 3 2
2
5 5 2 3 2 5 2
6 2
2 3 2
7 7 2 7 2 3 2
8 2
2
2 3
9 3 2
2
2
A 2 3 2 5 2 5
B
2 3 2
2
C 2
2 3 2
D
2
2 3 2
E 2 7 2 7 2 3
F 3 2 5 2 5 2
G 2 3 2
2
H
2 3 2
2
I 2
2 3 2
J
2
2 3 2
K 2 5 2 5 2 3
L 3 2 7 2 7 2
M 2 3 2
2
N
2 3 2
2
P 2
2 3 2
Q 5 2 5 2 3 2
R 2
2
2 3
S 3 2
2
2
T 2 3 2 7 2 7
U
2 3 2
2
V 2 5 2 3 2 5
W
2
2 3 2
X 2
2
2 3
Y 3 2
2
2
Z 2 3 2
2
Possible primes 8 8 8 8 8 8

What we see is that whilst there can be ten primes in a span of thirty-five numbers, there can as shown above never be more than eight primes between consecutive multiples of 35. The first occurrence of eight primes between multiples of 35 is between the pentatrigesimal numbers 4L0 and 4LZ, or in decimal 5,635 to 5,669, whereby 5,639, 5,641, 5,647, 5,651, 5,653, 5,657, 5,659 and 5,669 are all primes.

What I found reading today is that OEIS has actually tabulated the number of possible primes between multiples of base b, as well as the number possible in an arbitrary sequence of length n. I have compared “decades” and “centuries” in all bases up to 35, with sequences serving “centuries” (k00) shaded in grey:

Base Arbitrary Multiple Difference
2 1 1 0
3 2 1 -1
2 4 2 2 0
5 2 2 0
6 2 2 0
7 3 2 -1
8 3 3 0
3 9 4 3 -1
10 4 4 0
11 4 4 0
12 4 4 0
13 5 4 -1
14 5 4 -1
15 5 4 -1
4 16 5 5 0
17 6 5 -1
18 6 6 0
19 6 6 0
20 6 6 0
21 7 5 -2
22 7 7 0
23 7 7 0
24 7 6 -1
5 25 7 7 0
26 7 7 0
27 8 7 -1
28 8 7 -1
29 8 8 0
30 8 7 -1
31 9 8 -1
32 9 9 0
33 10 8 -2
34 10 10 0
35 10 8 -2
6 36 10 10 0
7 49 13 12 -1
8 64 16 16 0
9 81 20 19 -1
10 100 23 23 0
11 121 27 26 -1
12 144 31 30 -1
13 169 37 36 -1
14 196 41 41 0
15 225 46 45 -1
17 289 58 56 -2
18 324 62 62 0
19 361 68 67 -1
20 400 75 75 0
21 441 81 80 -1
22 484 88 87 -1
23 529 95 94 -1
24 576 101 101 0
25 625 109 109 0
26 676 117 116 -1
27 729 124 124 0
28 784 132 131 -1
29 841 141 139 -2
30 900 149 149 0
31 961 158 156 -2

As you can see, the pentatrigesimal case I discovered earlier is not exceptional. For base 21, one already sees that there can never be more than five primes between multiples of 21, but there can be seven primes within arbitrary sequences of 21 numbers.

Nonetheless, the list is revealing because — as it did for me — it gave a false impression that there must be more prime-dense “decades” in pentatrigesimal than that from 4L0 to 4LZ (5,635 to 5,669 in decimal). In cases like this one does need to take some care looking for prime constellations that can actually be easily proved impossible.

Tuesday, 29 August 2023

Illustrating each count of composites not divisible by 2, 3, or 5

Following on from my previous post, I will now illustrate each count of composite numbers no divisible by 2, 3, or 5 in a century, except for those not known to occur.

Some of these tables have been shown on this blog at ‘The 16,719th century and other prime-free sequences’ from almost a decade ago.

For a few counts affected by the “eight-digit gap” — those from nine to eleven — I will show those centuries immediately below and above the “eight-digit gap” in a subsequent table for more perspective. Centuries with fewer than nine composites not divisible by 2, 3, or 5 also show the effect of the “eight-digit gap”, but this is obscured because:
  1. no century below the eight-digit gap has six or eight composites not divisible by 2, 3, or 5
  2. although with seven such composites at least one larger than the second century is known, it is unlikely to be the second smallest
  3. none with the theoretically possible four or five composites not divisible by 2, 3, or 5 have yet been found.
n Factorisations of numbers not divisible by 2, 3, or 5
3 7 is prime
11 is prime
13 is prime
17 is prime
19 is prime
23 is prime
29 is prime
31 is prime
37 is prime
41 is prime
43 is prime
47 is prime
1. 49 = 7 × 7
53 is prime
59 is prime
61 is prime
67 is prime
71 is prime
73 is prime
2. 77 = 7 × 11
79 is prime
83 is prime
89 is prime
3. 91 = 7 × 13
97 is prime
4 (none known)
5 (none known)
6 39433867730216371575457664401 is prime
39433867730216371575457664407 is prime
39433867730216371575457664411 is prime
39433867730216371575457664413 is prime
39433867730216371575457664417 is prime
1. 39433867730216371575457664419 = 7 × 19 × 23 × 229 × 2179 × 25834328988233247551
39433867730216371575457664423 is prime
39433867730216371575457664429 is prime
39433867730216371575457664431 is prime
39433867730216371575457664437 is prime
39433867730216371575457664441 is prime
39433867730216371575457664443 is prime
2. 39433867730216371575457664447 = 7 × 11 × 29 × 993961 × 17766885682653834719
39433867730216371575457664449 is prime
39433867730216371575457664453 is prime
39433867730216371575457664459 is prime
3. 39433867730216371575457664461 = 7 × 13 × 233 × 6050207 × 307398559703912641
39433867730216371575457664467 is prime
39433867730216371575457664471 is prime
39433867730216371575457664473 is prime
39433867730216371575457664477 is prime
39433867730216371575457664479 is prime
39433867730216371575457664483 is prime
4. 39433867730216371575457664489 = 7 × 17 × 9470228338819 × 34991451936949
5. 39433867730216371575457664491 = 11 × 47 × 432569 × 331736687 × 531532637641
6. 39433867730216371575457664497 = 389 × 547968840241 × 184996669834253
7 101 is prime
103 is prime
107 is prime
109 is prime
113 is prime
1. 119 = 7 × 17
2. 121 = 11 × 11
127 is prime
131 is prime
3. 133 = 7 × 19
137 is prime
139 is prime
4. 143 = 11 × 13
149 is prime
151 is prime
157 is prime
5. 161 = 7 × 23
163 is prime
167 is prime
6. 169 = 13 × 13
173 is prime
179 is prime
181 is prime
7. 187 = 11 × 17
191 is prime
193 is prime
197 is prime
199 is prime
8 2335286971401803 is prime
2335286971401809 is prime
1. 2335286971401811 = 7 × 137 × 9433 × 258149813
2. 2335286971401817 = 37 × 89 × 257 × 1319 × 2092043
2335286971401821 is prime
2335286971401823 is prime
2335286971401827 is prime
2335286971401829 is prime
3. 2335286971401833 = 83 × 83 × 409 × 2153 × 384961
4. 2335286971401839 = 7 × 7 × 13 × 17 × 29 × 41 × 47 × 313 × 12329
2335286971401841 is prime
2335286971401847 is prime
2335286971401851 is prime
5. 2335286971401853 = 7 × 79 × 5413 × 26711 × 29207
6. 2335286971401857 = 11 × 127 × 1671644217181
2335286971401859 is prime
2335286971401863 is prime
2335286971401869 is prime
2335286971401871 is prime
2335286971401877 is prime
7. 2335286971401881 = 7 × 19 × 125507 × 139900951
2335286971401883 is prime
2335286971401887 is prime
2335286971401889 is prime
8. 2335286971401893 = 35255371 × 66239183
2335286971401899 is prime
9 1. 1403 = 23 × 61
1409 is prime
2. 1411 = 17 × 83
3. 1417 = 13 × 109
4. 1421 = 7 × 7 × 29
1423 is prime
1427 is prime
1429 is prime
1433 is prime
1439 is prime
5. 1441 = 11 × 131
1447 is prime
1451 is prime
1453 is prime
6. 1457 = 31 × 47
1459 is prime
7. 1463 = 7 × 11 × 19
8. 1469 = 13 × 113
1471 is prime
9. 1477 = 7 × 211
1481 is prime
1483 is prime
1487 is prime
1489 is prime
1493 is prime
1499 is prime
10 1. 203 = 7 × 29
2. 209 = 11 × 19
211 is prime
3. 217 = 7 × 31
4. 221 = 13 × 17
223 is prime
227 is prime
229 is prime
233 is prime
239 is prime
241 is prime
5. 247 = 13 × 19
251 is prime
6. 253 = 11 × 23
257 is prime
7. 259 = 7 × 37
263 is prime
269 is prime
271 is prime
277 is prime
281 is prime
283 is prime
8. 287 = 7 × 41
9. 289 = 17 × 17
293 is prime
10. 299 = 13 × 23
11 401 is prime
1. 403 = 13 × 31
2. 407 = 11 × 37
409 is prime
3. 413 = 7 × 59
419 is prime
421 is prime
4. 427 = 7 × 61
431 is prime
433 is prime
5. 437 = 19 × 23
439 is prime
443 is prime
449 is prime
6. 451 = 11 × 41
457 is prime
461 is prime
463 is prime
467 is prime
7. 469 = 7 × 67
8. 473 = 11 × 43
479 is prime
9. 481 = 13 × 37
487 is prime
491 is prime
10. 493 = 17 × 29
11. 497 = 7 × 71
499 is prime
12 503 is prime
509 is prime
1. 511 = 7 × 73
2. 517 = 11 × 47
521 is prime
523 is prime
3. 527 = 17 × 31
4. 529 = 23 × 23
5. 533 = 13 × 41
6. 539 = 7 × 7 × 11
541 is prime
547 is prime
7. 551 = 19 × 29
8. 553 = 7 × 79
557 is prime
9. 559 = 13 × 43
563 is prime
569 is prime
571 is prime
577 is prime
10. 581 = 7 × 83
11. 583 = 11 × 53
587 is prime
12. 589 = 19 × 31
593 is prime
599 is prime
13 1601 is prime
1. 1603 = 7 × 229
1607 is prime
1609 is prime
1613 is prime
1619 is prime
1621 is prime
1627 is prime
2. 1631 = 7 × 233
3. 1633 = 23 × 71
1637 is prime
4. 1639 = 11 × 149
5. 1643 = 31 × 53
6. 1649 = 17 × 97
7. 1651 = 13 × 127
1657 is prime
8. 1661 = 11 × 151
1663 is prime
1667 is prime
1669 is prime
9. 1673 = 7 × 239
10. 1679 = 23 × 73
11. 1681 = 41 × 41
12. 1687 = 7 × 241
13. 1691 = 19 × 89
1693 is prime
1697 is prime
1699 is prime
14 701 is prime
1. 703 = 19 × 37
2. 707 = 7 × 101
709 is prime
3. 713 = 23 × 31
719 is prime
4. 721 = 7 × 103
727 is prime
5. 731 = 17 × 43
733 is prime
6. 737 = 11 × 67
739 is prime
743 is prime
7. 749 = 7 × 107
751 is prime
757 is prime
761 is prime
8. 763 = 7 × 109
9. 767 = 13 × 59
769 is prime
773 is prime
10. 779 = 19 × 41
11. 781 = 11 × 71
787 is prime
12. 791 = 7 × 113
13. 793 = 13 × 61
797 is prime
14. 799 = 17 × 47
15 1901 is prime
1. 1903 = 11 × 173
1907 is prime
2. 1909 = 23 × 83
1913 is prime
3. 1919 = 19 × 101
4. 1921 = 17 × 113
5. 1927 = 41 × 47
1931 is prime
1933 is prime
6. 1937 = 13 × 149
7. 1939 = 7 × 277
8. 1943 = 29 × 67
1949 is prime
1951 is prime
9. 1957 = 19 × 103
10. 1961 = 37 × 53
11. 1963 = 13 × 151
12. 1967 = 7 × 281
13. 1969 = 11 × 179
1973 is prime
1979 is prime
14. 1981 = 7 × 283
1987 is prime
15. 1991 = 11 × 181
1993 is prime
1997 is prime
1999 is prime
16 1. 2101 = 11 × 191
2. 2107 = 7 × 7 × 43
2111 is prime
2113 is prime
3. 2117 = 29 × 73
4. 2119 = 13 × 163
5. 2123 = 11 × 193
2129 is prime
2131 is prime
2137 is prime
2141 is prime
2143 is prime
6. 2147 = 19 × 113
7. 2149 = 7 × 307
2153 is prime
8. 2159 = 17 × 127
2161 is prime
9. 2167 = 11 × 197
10. 2171 = 13 × 167
11. 2173 = 41 × 53
12. 2177 = 7 × 311
2179 is prime
13. 2183 = 37 × 59
14. 2189 = 11 × 199
15. 2191 = 7 × 313
16. 2197 = 13 × 13 × 13
17 1301 is prime
1303 is prime
1307 is prime
1. 1309 = 7 × 11 × 17
2. 1313 = 13 × 101
1319 is prime
1321 is prime
1327 is prime
3. 1331 = 11 × 11 × 11
4. 1333 = 31 × 43
5. 1337 = 7 × 191
6. 1339 = 13 × 103
7. 1343 = 17 × 79
8. 1349 = 19 × 71
9. 1351 = 7 × 193
10. 1357 = 23 × 59
1361 is prime
11. 1363 = 29 × 47
1367 is prime
12. 1369 = 37 × 37
1373 is prime
13. 1379 = 7 × 197
1381 is prime
14. 1387 = 19 × 73
15. 1391 = 13 × 107
16. 1393 = 7 × 199
17. 1397 = 11 × 127
1399 is prime
18 1. 3101 = 7 × 443
2. 3103 = 29 × 107
3. 3107 = 13 × 239
3109 is prime
4. 3113 = 11 × 283
3119 is prime
3121 is prime
5. 3127 = 53 × 59
6. 3131 = 31 × 101
7. 3133 = 13 × 241
3137 is prime
8. 3139 = 43 × 73
9. 3143 = 7 × 449
10. 3149 = 47 × 67
11. 3151 = 23 × 137
12. 3157 = 7 × 11 × 41
13. 3161 = 29 × 109
3163 is prime
3167 is prime
3169 is prime
14. 3173 = 19 × 167
15. 3179 = 11 × 17 × 17
3181 is prime
3187 is prime
3191 is prime
16. 3193 = 31 × 103
17. 3197 = 23 × 139
18. 3199 = 7 × 457
19 1. 4301 = 11 × 17 × 23
2. 4303 = 13 × 331
3. 4307 = 59 × 73
4. 4309 = 31 × 139
5. 4313 = 19 × 227
6. 4319 = 7 × 617
7. 4321 = 29 × 149
4327 is prime
8. 4331 = 61 × 71
9. 4333 = 7 × 619
4337 is prime
4339 is prime
10. 4343 = 43 × 101
4349 is prime
11. 4351 = 19 × 229
4357 is prime
12. 4361 = 7 × 7 × 89
4363 is prime
13. 4367 = 11 × 397
14. 4369 = 17 × 257
4373 is prime
15. 4379 = 29 × 151
16. 4381 = 13 × 337
17. 4387 = 41 × 107
4391 is prime
18. 4393 = 23 × 191
4397 is prime
19. 4399 = 53 × 83
20 1. 6401 = 37 × 173
2. 6403 = 19 × 337
3. 6407 = 43 × 149
4. 6409 = 13 × 17 × 29
5. 6413 = 11 × 11 × 53
6. 6419 = 7 × 7 × 131
6421 is prime
6427 is prime
7. 6431 = 59 × 109
8. 6433 = 7 × 919
9. 6437 = 41 × 157
10. 6439 = 47 × 137
11. 6443 = 17 × 379
6449 is prime
6451 is prime
12. 6457 = 11 × 587
13. 6461 = 7 × 13 × 71
14. 6463 = 23 × 281
15. 6467 = 29 × 223
6469 is prime
6473 is prime
16. 6479 = 11 × 19 × 31
6481 is prime
17. 6487 = 13 × 499
6491 is prime
18. 6493 = 43 × 151
19. 6497 = 73 × 89
20. 6499 = 67 × 97
21 1. 14201 = 11 × 1291
2. 14203 = 7 × 2029
14207 is prime
3. 14209 = 13 × 1093
4. 14213 = 61 × 233
5. 14219 = 59 × 241
14221 is prime
6. 14227 = 41 × 347
7. 14231 = 7 × 19 × 107
8. 14233 = 43 × 331
9. 14237 = 23 × 619
10. 14239 = 29 × 491
14243 is prime
14249 is prime
14251 is prime
11. 14257 = 53 × 269
12. 14261 = 13 × 1097
13. 14263 = 17 × 839
14. 14267 = 11 × 1297
15. 14269 = 19 × 751
16. 14273 = 7 × 2039
17. 14279 = 109 × 131
14281 is prime
18. 14287 = 7 × 13 × 157
19. 14291 = 31 × 461
14293 is prime
20. 14297 = 17 × 29 × 29
21. 14299 = 79 × 181
22 1. 31403 = 31 × 1013
2. 31409 = 7 × 7 × 641
3. 31411 = 101 × 311
4. 31417 = 89 × 353
5. 31421 = 13 × 2417
6. 31423 = 7 × 67 × 67
7. 31427 = 11 × 2857
8. 31429 = 53 × 593
9. 31433 = 17 × 43 × 43
10. 31439 = 149 × 211
11. 31441 = 23 × 1367
12. 31447 = 13 × 41 × 59
13. 31451 = 7 × 4493
14. 31453 = 71 × 443
15. 31457 = 83 × 379
16. 31459 = 163 × 193
17. 31463 = 73 × 431
31469 is prime
18. 31471 = 11 × 2861
31477 is prime
31481 is prime
19. 31483 = 19 × 1657
20. 31487 = 23 × 37 × 37
31489 is prime
21. 31493 = 7 × 11 × 409
22. 31499 = 13 × 2423
23 1. 40301 = 191 × 211
2. 40303 = 41 × 983
3. 40307 = 17 × 2371
4. 40309 = 173 × 233
5. 40313 = 7 × 13 × 443
6. 40319 = 23 × 1753
7. 40321 = 61 × 661
8. 40327 = 7 × 7 × 823
9. 40331 = 31 × 1301
10. 40333 = 53 × 761
11. 40337 = 11 × 19 × 193
12. 40339 = 13 × 29 × 107
40343 is prime
13. 40349 = 157 × 257
40351 is prime
40357 is prime
40361 is prime
14. 40363 = 181 × 223
15. 40367 = 37 × 1091
16. 40369 = 7 × 73 × 79
17. 40373 = 47 × 859
18. 40379 = 149 × 271
19. 40381 = 11 × 3671
40387 is prime
20. 40391 = 13 × 13 × 239
21. 40393 = 31 × 1303
22. 40397 = 7 × 29 × 199
23. 40399 = 71 × 569
24 1. 180101 = 19 × 9479
2. 180103 = 7 × 11 × 2339
3. 180107 = 389 × 463
4. 180109 = 233 × 773
5. 180113 = 23 × 41 × 191
6. 180119 = 29 × 6211
7. 180121 = 281 × 641
8. 180127 = 43 × 59 × 71
9. 180131 = 7 × 25733
10. 180133 = 61 × 2953
180137 is prime
11. 180139 = 19 × 19 × 499
12. 180143 = 151 × 1193
13. 180149 = 17 × 10597
14. 180151 = 47 × 3833
15. 180157 = 257 × 701
180161 is prime
16. 180163 = 67 × 2689
17. 180167 = 13 × 13859
18. 180169 = 11 × 11 × 1489
19. 180173 = 7 × 7 × 3677
180179 is prime
180181 is prime
20. 180187 = 7 × 25741
21. 180191 = 11 × 16381
22. 180193 = 13 × 83 × 167
23. 180197 = 367 × 491
24. 180199 = 79 × 2281
25 1. 155903 = 11 × 14173
2. 155909 = 13 × 67 × 179
3. 155911 = 7 × 22273
4. 155917 = 23 × 6779
155921 is prime
5. 155923 = 41 × 3803
6. 155927 = 241 × 647
7. 155929 = 211 × 739
8. 155933 = 19 × 29 × 283
9. 155939 = 7 × 22277
10 .155941 = 17 × 9173
11. 155947 = 11 × 14177
12. 155951 = 277 × 563
13. 155953 = 7 × 22279
14. 155957 = 83 × 1879
15. 155959 = 263 × 593
16. 155963 = 23 × 6781
17. 155969 = 11 × 11 × 1289
18. 155971 = 19 × 8209
19. 155977 = 61 × 2557
20. 155981 = 7 × 22283
21. 155983 = 151 × 1033
22. 155987 = 13 × 13 × 13 × 71
23. 155989 = 389 × 401
24. 155993 = 47 × 3319
25. 155999 = 257 × 607
26 1. 370901 = 421 × 881
2. 370903 = 13 × 103 × 277
3. 370907 = 167 × 2221
4. 370909 = 7 × 11 × 4817
5. 370913 = 73 × 5081
370919 is prime
6. 370921 = 23 × 16127
7. 370927 = 41 × 83 × 109
8. 370931 = 11 × 33721
9. 370933 = 59 × 6287
10. 370937 = 7 × 19 × 2789
11. 370939 = 29 × 12791
12. 370943 = 347 × 1069
370949 is prime
13. 370951 = 7 × 197 × 269
14. 370957 = 17 × 21821
15. 370961 = 43 × 8627
16. 370963 = 409 × 907
17. 370967 = 23 × 127 × 127
18. 370969 = 107 × 3467
19. 370973 = 101 × 3673
20. 370979 = 7 × 7 × 67 × 113
21. 370981 = 13 × 28537
22. 370987 = 349 × 1063
23. 370991 = 17 × 139 × 157
24. 370993 = 7 × 52999
25. 370997 = 11 × 29 × 1163
26. 370999 = 37 × 37 × 271
27 1. 268301 = 11 × 24391
2. 268303 = 7 × 38329
3. 268307 = 13 × 20639
4. 268309 = 71 × 3779
5. 268313 = 157 × 1709
6. 268319 = 251 × 1069
7. 268321 = 179 × 1499
8. 268327 = 151 × 1777
9. 268331 = 7 × 38333
10. 268333 = 13 × 20641
11. 268337 = 19 × 29 × 487
12. 268339 = 53 × 61 × 83
268343 is prime
13. 268349 = 149 × 1801
14. 268351 = 127 × 2113
15. 268357 = 101 × 2657
16. 268361 = 37 × 7253
17. 268363 = 43 × 79 × 79
18. 268367 = 11 × 31 × 787
19. 268369 = 167 × 1607
20. 268373 = 7 × 7 × 5477
21. 268379 = 17 × 15787
22. 268381 = 349 × 769
23. 268387 = 7 × 23 × 1667
24. 268391 = 59 × 4549
25. 268393 = 311 × 863
26. 268397 = 239 × 1123
27. 268399 = 97 × 2767
28 1. 6085001 = 13 × 223 × 2099
2. 6085003 = 139 × 43777
3. 6085007 = 67 × 90821
4. 6085009 = 7 × 241 × 3607
5. 6085013 = 11 × 641 × 863
6. 6085019 = 89 × 68371
7. 6085021 = 31 × 196291
8. 6085027 = 13 × 468079
9. 6085031 = 17 × 263 × 1361
10. 6085033 = 313 × 19441
11. 6085037 = 7 × 869291
12. 6085039 = 181 × 33619
13. 6085043 = 47 × 129469
14. 6085049 = 269 × 22621
15. 6085051 = 7 × 869293
16. 6085057 = 11 × 37 × 14951
17. 6085061 = 73 × 83357
18. 6085063 = 907 × 6709
19. 6085067 = 881 × 6907
20. 6085069 = 191 × 31859
21. 6085073 = 19 × 320267
22. 6085079 = 7 × 11 × 13 × 6079
23. 6085081 = 131 × 46451
24. 6085087 = 23 × 23 × 11503
25. 6085091 = 1013 × 6007
26. 6085093 = 7 × 869299
27. 6085097 = 41 × 193 × 769
28. 6085099 = 17 × 29 × 12343

Surrounding Centuries Not Shown for Composite Counts Affected by Eight-Digit Gap

n Factorisations of numbers not divisible by 2, 3, or 5
7

12373565131489557507901 is prime
12373565131489557507907 is prime
1. 12373565131489557507911 = 7 × 41 × 158429 × 272131158797357
12373565131489557507913 is prime
12373565131489557507917 is prime
2. 12373565131489557507919 = 11 × 11 × 11 × 10457 × 14533 × 61172243929
12373565131489557507923 is prime
12373565131489557507929 is prime
12373565131489557507931 is prime
12373565131489557507937 is prime
3. 12373565131489557507941 = 11 × 13 × 104551 × 827619319690237
12373565131489557507943 is prime
12373565131489557507947 is prime
12373565131489557507949 is prime
4. 12373565131489557507953 = 7 × 64891 × 367501 × 74123141369
12373565131489557507959 is prime
12373565131489557507961 is prime
5. 12373565131489557507967 = 7 × 13 × 17 × 43 × 5531 × 87107 × 386081831
12373565131489557507971 is prime
12373565131489557507973 is prime
6. 12373565131489557507977 = 10141 × 1220152364805202397
12373565131489557507979 is prime
12373565131489557507983 is prime
12373565131489557507989 is prime
12373565131489557507991 is prime
7. 12373565131489557507997 = 29 × 1061 × 402143882852532013

9 190818931155803 is prime
1. 190818931155809 = 11 × 43 × 54851 × 7354883
2. 190818931155811 = 7 × 7 × 3894263901139
190818931155817 is prime
190818931155821 is prime
190818931155823 is prime
3. 190818931155827 = 13 × 1259 × 11658760381
4. 190818931155829 = 37 × 983 × 5246458199
190818931155833 is prime
5. 190818931155839 = 7 × 17 × 19 × 887 × 95147477
190818931155841 is prime
6. 190818931155847 = 59 × 73 × 9883 × 4482887
190818931155851 is prime
7. 190818931155853 = 7 × 11 × 13 × 499 × 382020647
190818931155857 is prime
190818931155859 is prime
190818931155863 is prime
190818931155869 is prime
190818931155871 is prime
8. 190818931155877 = 19 × 1571 × 56533 × 113081
9. 190818931155881 = 7 × 23 × 84509 × 14024669
190818931155883 is prime
190818931155887 is prime
190818931155889 is prime
190818931155893 is prime
190818931155899 is prime
10 23003 is prime
1. 23009 = 7 × 19 × 173
23011 is prime
23017 is prime
23021 is prime
2. 23023 = 7 × 11 × 13 × 23
23027 is prime
23029 is prime
3. 23033 = 31 × 743
23039 is prime
23041 is prime
4. 23047 = 19 × 1213
5. 23051 = 7 × 37 × 89
23053 is prime
23057 is prime
23059 is prime
23063 is prime
6. 23069 = 17 × 23 × 59
23071 is prime
7. 23077 = 47 × 491
23081 is prime
8. 23083 = 41 × 563
23087 is prime
9. 23089 = 11 × 2099
10. 23093 = 7 × 3299
23099 is prime
1. 2466269101 = 17 × 61 × 2378273
2466269107 is prime
2466269111 is prime
2. 2466269113 = 7 × 11 × 1201 × 26669
3. 2466269117 = 13 × 47 × 971 × 4157
2466269119 is prime
2466269123 is prime
4. 2466269129 = 67 × 2269 × 16223
2466269131 is prime
2466269137 is prime
5. 2466269141 = 7 × 19 × 18543377
6. 2466269143 = 13 × 173 × 541 × 2027
2466269147 is prime
2466269149 is prime
2466269153 is prime
2466269159 is prime
2466269161 is prime
2466269167 is prime
7. 2466269171 = 43 × 57355097
2466269173 is prime
2466269177 is prime
8. 2466269179 = 11 × 19 × 631 × 18701
9. 2466269183 = 7 × 103 × 113 × 30271
2466269189 is prime
2466269191 is prime
10. 2466269197 = 7 × 157 × 829 × 2707
11 1. 2967301 = 43 × 151 × 457
2. 2967307 = 7 × 109 × 3889
3. 2967311 = 53 × 55987
4. 2967313 = 719 × 4127
2967317 is prime
5. 2967319 = 79 × 37561
2967323 is prime
2967329 is prime
2967331 is prime
2967337 is prime
6. 2967341 = 13 × 228257
2967343 is prime
2967347 is prime
7. 2967349 = 7 × 11 × 89 × 433
2967353 is prime
2967359 is prime
2967361 is prime
8. 2967367 = 13 × 17 × 29 × 463
9. 2967371 = 11 × 269761
2967373 is prime
10. 2967377 = 7 × 73 × 5807
2967379 is prime
2967383 is prime
2967389 is prime
11. 2967391 = 7 × 7 × 23 × 2633
2967397 is prime
2967401 = 17 × 19 × 9187
2967403 is prime
2967407 is prime
2967409 is prime
1. 98195303 = 23 × 421 × 10141
98195309 is prime
98195311 is prime
2. 98195317 = 11 × 1621 × 5507
3. 98195321 = 7 × 31 × 149 × 3037
98195323 is prime
98195327 is prime
98195329 is prime
98195333 is prime
4. 98195339 = 11 × 977 × 9137
5. 98195341 = 79 × 1 242979
98195347 is prime
98195351 is prime
98195353 is prime
6. 98195357 = 13 × 1511 × 4999
7. 98195359 = 43 × 103 × 22171
8. 98195363 = 7 × 7 × 19 × 29 × 3637
9. 98195369 = 41 × 2395009
98195371 is prime
10. 98195377 = 7 × 947 × 14813
98195381 is prime
11. 98195383 = 11 × 13 × 17 × 31 × 1303
98195387 is prime
98195389 is prime
98195393 is prime
98195399 is prime
The 123,735,651,314,895,575,080th century is unlikely to be the second-smallest century with only seven composite numbers not divisible by 2, 3, or 5. It is however the second one known, so I have included it.

In the case of the 29,674th century, I have included the first decade of the following century because of the remarkable prime richness of the 100 numbers ending with 2,967,410. The eighteen primes in those 100 numbers is not equalled between 113,180 and 341,078,531,679 (that is, greater than 341 billion, or over one hundred thousand times larger in magnitude than the 29,674th century), and no consecutive multiples of 100 between 200 and 122,853,771,370,899 contain as many as eighteen primes between them. Moreover, as Brian Kehrig noted in correspondence with me, seven rather than eight is the number of digits for which there should be fewest centuries or sequences exceptionally rich in prime numbers, making the sequence from 2,967,310 to 2,967,410 even more exceptional.

The century from 98,195,300 to 98,195,399 is still within the eight-digit gap — after all, it contains eight digits and not more or less — but the eleven composites not divisible by 2, 3, or 5 is the fewest since the 29,674th century and shows one moving away from the “core” of that gap. There are four centuries between 98,195,400 and 839,296,299 containing only eleven composites not divisible by 2, 3, or 5, but it would be superfluous to include all of them and the 981,954th century should be reasonably illustrative.