“Full digits” in sixteen-prime centuries

In my previous post, I noted that I was trying to see how many of the smallest sixteen-prime centuries had seven primes ending in one digit. This latter phenomenon is one I have only noticed recently and is not recorded on OEIS at all.

Working through the first one thousand centuries with sixteen primes on OEIS (sequence A186408) I found the following cases:

nth century with 16 primes	k	n mod 21	Seven primes ending in
23	1097546872	16	9
223	926471559529	19	3
255	1194384398740	1	7
289	1534136303215	19	3
294	1553991849772	19	3
307	1644652079911	1	7
315	1725961694479	1	7
482	3627907603807	4	1
750	8673617675503	4	1
883	11557194277084	1	7
925	12714434652190	19	3

The table shows more cases of seven primes ending in 3 or 7 than on 1 or 9. This is because, as noted in the previous post, there is a very strong preponderance of sixteen-prime centuries that are either 1 or 19 modulo 21. These two moduli 21 account for over half of the first thousand sixteen-prime centuries above the eight-digit gap. As you can see, a 1 modulo 21 century may have seven primes ending in 7 (but no other digit) and a 19 modulo 21 century may have seven primes ending in 3 but no other digit. A couple of year ago I calculated that the probability of a 16-prime century having seven primes ending in the same digit was about 1 in 100. The eleven centuries above agrees fairly well with this expectation.

I have not checked seventeen- or eighteen- prime centuries — this is tough to do because Excel cannot read numbers beyond 10¹⁵ to the precise whole number — but the fact that the first seventeen-prime century with seven primes ending in one digit has seven primes ending in 7 does suggest a similar pattern.

The full factorisations of the smallest century with seven primes ending in each digit, with primes formign the group of seven coloured in red:

• 109754687201 = 641 × 1249 × 137089
• 109754687203 = 7 × 439 × 35 715811
• 109754687207 = 11 × 17 × 83 × 7071367
• 109754687209 is prime
• 109754687213 is prime
• 109754687219 is prime
• 109754687221 is prime
• 109754687227 = 499 × 947 × 232259
• 109754687231 = 7 × 41 × 827 × 462419
• 109754687233 is prime
• 109754687237 = 13 × 89 × 97 × 157 × 6229
• 109754687239 is prime
• 109754687243 is prime
• 109754687249 is prime
• 109754687251 = 113 × 82460321
• 109754687257 is prime
• 109754687261 is prime
• 109754687263 = 13 × 37 × 228180223
• 109754687267 is prime
• 109754687269 is prime
• 109754687273 = 74 × 11 × 31 × 134053
• 109754687279 is prime
• 109754687281 = 29 × 79 × 47906891
• 109754687287 = 7 × 67 × 107 × 239 × 9151
• 109754687291 = 19 × 73 × 79130993
• 109754687293 is prime
• 109754687297 is prime
• 109754687299 is prime
• 92647155952901 = 11 × 673 × 1229 × 10182923
• 92647155952903 is prime
• 92647155952907 is prime
• 92647155952909 is prime
• 92647155952913 is prime
• 92647155952919 = 41 × 3167 × 713510177
• 92647155952921 = 17 × 863 × 48397 × 130483
• 92647155952927 is prime
• 92647155952931 = 24953 × 3712866427
• 92647155952933 is prime
• 92647155952937 is prime
• 92647155952939 = 7 × 73 × 181305588949
• 92647155952943 is prime
• 92647155952949 = 13 × 79 × 32099 × 2810413
• 92647155952951 = 29 × 229 × 13950783911
• 92647155952957 = 186187 × 497602711
• 92647155952961 is prime
• 92647155952963 is prime
• 92647155952967 = 7 × 11 × 79843 × 15069697
• 92647155952969 is prime
• 92647155952973 is prime
• 92647155952979 is prime
• 92647155952981 = 7 × 23 × 575448173621
• 92647155952987 is prime
• 92647155952991 = 19 × 97 × 50269753637
• 92647155952993 is prime
• 92647155952997 is prime 
• 92647155952999 = 53 × 647 × 10133 × 266633
• 119438439874001 = 19 × 2425019 × 2592241
• 119438439874003 = 29 × 4118566892207 
• 119438439874007 is prime
• 119438439874009 is prime
• 119438439874013 = 13 × 53 × 79 × 1787 × 1227929
• 119438439874019 = 7 × 1907 × 13687 × 653713
• 119438439874021 is prime 
• 119438439874027 is prime
• 119438439874031 = 23 × 257 × 20206130921
• 119438439874033 = 7 × 161233 × 105825943
• 119438439874037 is prime
• 119438439874039 = 11 × 13 × 19 × 149 × 211 × 613 × 2281
• 119438439874043 = 1473853 × 81038231
• 119438439874049 is prime
• 119438439874051 is prime
• 119438439874057 is prime
• 119438439874061 = 7 × 11 × 29 × 97 × 6863 × 80347
• 119438439874063 is prime
• 119438439874067 is prime
• 119438439874069 is prime
• 119438439874073 = 1221083 × 97813531
• 119438439874079 is prime
• 119438439874081 is prime
• 119438439874087 is prime
• 119438439874091 = 13 × 37 × 59 × 18341 × 229469
• 119438439874093 = 17 × 89459 × 78536431
• 119438439874097 is prime
• 119438439874099 is prime
• 362790760380701 is prime
• 362790760380703 is prime
• 362790760380707 = 17 × 19 × 109 × 10304506501
• 362790760380709 is prime
• 362790760380713 = 7 × 51 827251 482959
• 362790760380719 = 11 × 32980978216429
• 362790760380721 is prime
• 362790760380727 = 7 × 31 × 193 × 16889 × 512903
• 362790760380731 is prime
• 362790760380733 = 1759 × 206248300387
• 362790760380737 = 13 × 43 × 53 × 71 × 172 468661
• 362790760380739 is prime
• 362790760380743 = 433 × 837853950071
• 362790760380749 is prime
• 362790760380751 is prime
• 362790760380757 is prime
• 362790760380761 is prime
• 362790760380763 = 11 × 13 × 13 × 59 × 641 × 947 × 5449
• 362790760380767 is prime
• 362790760380769 = 7 × 23 × 47 × 401 × 557 × 214651
• 362790760380773 is prime
• 362790760380779 = 37 × 61 × 509 × 11783 × 26801
• 362790760380781 is prime
• 362790760380787 is prime
• 362790760380791 is prime
• 362790760380793 is prime
• 362790760380797 = 7 × 51827251482971
• 362790760380799 = 41 × 631 × 14023066769

Revealing the eight-digit gap again

In previous posts on prime numbers, I have noticed the existence of an “eight-digit gap” — a range of magnitudes within which the maximum number of primes in a century reaches a minus, and extending from about three million to 840 million.

Today, I attempted to see how many of the first 1,000 centuries with sixteen prime numbers, nine of which lie below the eight-digit gap, contained seven primes ending in one digit. As noted here, the first centuries with seven primes ending in same digit lie in the absolute core of the eight-digit gap, but I wanted to see just how common they are among sixteen-prime centuries larger than that gap. I previously did very brief studies for seventeen- and eighteen-prime centuries, and found that the sixteenth century with eighteen primes from 140,326,343,186,616,700 to 140,326,343,186,616,799 had seven primes ending in 1, but not seventeen-prime century had seven primes ending in any digit until the 179^th such century from 24,738,663,087,001,600 to 24,738,663,087,001,699 with seven primes ending in 7.

However, when working out possible cases of seven primes ending in the same digit, I noticed more clearly that the frequency of moduli 21 of sixteen-prime centuries larger than the eight-digit gap was in no way random. (Moduli 21 can exclude any century having seven primes ending in all or all but one of the four digits in which a multi-digit prime may terminate).

Number of First 1,000 Sixteen-Prime Centuries n modulo 21:

All centuries			Centuries above eight-digit gap			Centuries below eight-digit gap (includes 17- and 21-prime centuries)
n mod 21	total	percent	n mod 21	total	percent	n mod 21	total	percent
0	4	0.40%	0	3	0.30%	0	1	7.14%
1	276	27.60%	1	276	27.85%	1	2	14.29%
2	9	0.90%	2	8	0.81%	2	1	7.14%
3	16	1.60%	3	15	1.51%	3	1	7.14%
4	58	5.80%	4	58	5.85%	4	2	14.29%
5	7	0.70%	5	6	0.61%	5	1	7.14%
6	20	2.00%	6	19	1.92%	6	1	7.14%
7	28	2.80%	7	28	2.83%	7	0	0.00%
8	33	3.30%	8	33	3.33%	8	0	0.00%
9	7	0.70%	9	7	0.71%	9	0	0.00%
10	68	6.80%	10	66	6.66%	10	2	14.29%
11	14	1.40%	11	14	1.41%	11	0	0.00%
12	27	2.70%	12	27	2.72%	12	0	0.00%
13	33	3.30%	13	33	3.33%	13	0	0.00%
14	31	3.10%	14	31	3.13%	14	1	7.14%
15	8	0.80%	15	8	0.81%	15	0	0.00%
16	48	4.80%	16	47	4.74%	16	1	7.14%
17	11	1.10%	17	11	1.11%	17	0	0.00%
18	9	0.90%	18	9	0.91%	18	0	0.00%
19	282	28.20%	19	282	28.46%	19	0	0.00%
20	11	1.10%	20	10	1.01%	20	1	7.14%

What the table shows is that:

1. above the eight-digit gap there is a systematic variation in the frequencies of moduli 21 amongst sixteen-prime centuries, with extreme cases shaded
2. the fourteen centuries below the eight-digit gap with sixteen or more primes do not appear to follow this pattern
3. this suggests that there is a systematic character to prime-dense centuries above the eight-digit gap absent below that range of magnitudes

There is logic behind this pattern in that small prime factors have to combine in a few numbers with many factors to produce prime-rich centuries above (and within) the eight-digit gap. (As I note here, they also have to combine to produce large prime-free sequences, and this becomes harder as numbers get bigger because it is difficult for enough factors to “synchronise”). Below the eight-digit gap, there are fewer possible factors so there is less need to combine, and thus prime-rich centuries are less likely to fit into repetitive large groups of primes.

Least “average” yet most “American” — not the contradiction it seems

Several years ago, I read on Wikipedia about how Illinois has long been considered a microcosm of the United States. I accepted that ideal easily, given that the state — excluding the Memphis, Tennessee-aligned southernmost three counties of Alexander, Pulaski and Massac — is a mixture of the Midwest and the nonplantation South, which I have come to view as the most generically “American” regions of the country.

Upon actually reading the article, originally published in The Southern Illinoisian in 2007, I found something somewhat surprising:

“West Virginia was the least typical state — poorer, whiter, more rural — followed by Mississippi, New Hampshire, Vermont and Kentucky.”

That New Hampshire and Vermont are among the least typical states is impossible to question. Although certain features of their cultures — among the most distinctive in the US — may be retained from or linked to Puritan history, their dependence on highly global industries like tourism and finance exposes them to highly modern influences.

Mississippi is a slightly complicated case. Although ever since the Civil rights era it has been thought exceptional, in reality its highly trouble civil rights history is not “Mississippi exceptionalism” but “Mississippi genericism” at least vis-à-vis the other wholly “Deep South” states of South Carolina and Louisiana. Both South Carolina and Louisiana are among the most distinctive states of the US culturally and in political structures. Mississippi, contrariwise, had and has political structures analogous to the nonplantation South (and most of the US).

It nonetheless surprised me that Kentucky and West Virginia would be called the least typical states. In many ways, especially regarding culture — not a criterion used by The Southern Illinoisan — Appalachia and other nonplantation areas of the South comprise the most distinctly “American” region of the United States. In that sense, West Virginia and Kentucky are (two of) the most “American” states in the country.

The interesting thing is that, when I think about it, there is no contradiction between being the least “typical” state and the most “American” one. The reason Appalachia and other nonplantation regions of the South are such is that they are almost completely unexposed to outside cultural influences. Instead, they have become what James Löwen called “white ghettoes”, but which he and I agreed are more accurately named “white cloisters”. For historical reasons, many rural areas in the nonplantation South, the Midwest and the interior west have chosen to isolate themselves from outside culture for either religious or racial reasons — which of course may be linked.

1980s West Indies versus 1980s VFL?

Eleven years ago, I reacted in what I now recognise as a ridiculous manner to the discovery of an article in the Sydney Morning Herald that discussed the losses suffered by the 1984 Australian cricket tour of the West Indies. Having during the 1990s discovered an apparent pervasive anticorrelation between fast bowling strength and the profitability of first-class cricket, I long assumed the West Indies were an exception. Thus, my discovery that tours there were losing money made me emphasise beliefs established as a teenager, though others around me denied their truth.

Absence of evidence re the profitability of Caribbean tours before Lloyd worked out that spin was an unnecessary luxury precluded refutation of the argument that their extreme fast bowling strength caused the financial losses. Even people outside my family were unable to offer an answer with which I was satisfied — and they themselves never discussed the profitability of previous Caribbean tours.

Today when discussing the issue of how 1980s Wisdens overemphasised how bad England was and did not discuss what the West Indies were doing well, I received an interesting idea. This was that the financial losses of West Indian cricket during the 1980s were due to the increasing financial demands of their star players.

I had never thought of this argument before, but it is eerily similar to the situation faced by [Australian Rules] football during exactly the same era. With the VFL’s zoning and clearance rules declared illegal, and clubs in an era of inflation seeking to gain top players at any price, player costs continued to rise even when governments put the brakes on inflation, with the result that real wages of players increased at an increasing rate. Consequently, declining attendances as more and more people relocated to suburbs remote from the public transport essential to move large numbers of people to grounds produced much heavier losses than occasional attendance declines in previous eras. Clubs could not cut expenses as they previously could because of the demands players were making. These demands were further increased by a greater demand to win at any price (at least by most top football clubs). At the same time ruling classes who formerly patronised football turned to basketball, which was very easy to televise to people remote from public transport. The VFA, the WAFL, and lower leagues suffered even more than did the VFL. Ultimately, football had to reform on basketball’s terms — rationalised grounds and standardised conditions that made the game much easier to televise.

1980s West Indian cricket managers, no doubt, would have liked to reduce payments to star players to lower costs, but the Sydney Morning Herald implied that they knew this to be impossible. Also, exactly like [Australian Rules] football, cricket in the West Indies was struggling to compete with the rapidly growing National Basketball Association — based much closer than the MCC or Kerry Packer. Another similarity was that the West Indies were taking cricket to all grounds although only Queen’s Park in Trinidad was economically profitable, while football was played on unprofitable suburban ovals on which league-fixed ticket prices meant more popular clubs could not generate sufficient revenue from home games to improve facilities. The ruling classes of the West Indies, increasingly linked to the United States, had little interest in patronising better and cheaper accommodation at cricket venues.

A major difference is that the VFL by 1984 recognised for exactly this reason that suburban grounds needed to be phased out, whereas the West Indies Cricket Board was expanding first-class cricket to these very loss-making venues. Nonetheless, the similarities are both interesting and surprising — increasing demands from players and a shift of ruling class patrons to basketball are almost certainly behind the heavy losses of all [Australian Rules] football competitions in the 1980s and may well be behind the financial losses suffered by the West Indies while it dominated world cricket on the field.

Stupid computers or not the point?

When I studied in Melbourne University, I was outraged at the rude graffiti I discovered on virtually every toilet wall. Almost all if it was sexual in nature, and I always felt it encouraged violence towards women or towards men perceived as not strong enough. This latter tendency was aided by the fact that as a university student at the end of the 1990s I had strong memories of bullying both at school and on the street.

Coarse (or violent) language on films or in music seemed to me like a natural culprit for bullying behaviour. When I first heard of film ratings, I presumed their purpose must be safety — preventing children learning that violence is acceptable behaviour or naïvely thinking thus.

At the same time, I was learning to use Microsoft Word, and one function I quickly discovered was AutoCorrect. I quite quickly began to check spelling on Word, and soon discovered that I could add to and delete from AutoCorrect. Given my dislike of rude language, it felt entirely natural to add to AutoCorrect any rude word whose meaning in less coarse language was known. I did this a lot for a while, often repeatedly because my additions in the Melbourne University computer laboratories were probably not retained when the computers shut down at the end of each day.

All seemingly went well for a while, until I made, merely testing, a discovery that was truly shocking — that AutoCorrect was not case sensitive! This meant that “Dick” was corrected even when intended as someone’s name (I intended and assumed it would only correct when uncapitalised, but was shocked when “Dick Tyldesley”, a former Lancashire bowler, became “Penis Tyldesley”!). Following this, I assumed that the inability of AutoCorrect to be case sensitive meant computers really were not nearly so intelligent as everybody presumed. I consistently laughed at how something so specialised as a computer could be unable to distinguish capitalised from uncapitalised words and (in this case) correct only the uncapitalised. At the same time my brother said, critically but not aggressively, that by adding swear words I was turning AutoCorrect into “AutoCensor”. “AutoCensor” remains a really funny joke, much less gratuitous than the renaming of people called “Dick” by an addition intended only to AutoCorrect with a small initial letter.

After a while, the contrived nature of my additions to AutoCorrect made me think my original idea was silly because it was so difficult a job to accurately rewrite rude words as something less nasty and more often than not grauitously violent.

Then, I was told rather quietly one day by a university official than I had been banned from the computer labs for “tampering with AutoCorrect”. I was told that my tampering with AutoCorrect had ruined some other student’s essays — completely changing texts in such a way that they could not be mended. Unlike later cases at RMIT where I reacted extremely violently and angrily, I accepted this punishment because I knew very clearly that I had been altering AutoCorrect. Even if I felt my intentions were good, I had already realised that tampering with AutoCorrect simply could not do what I wanted it to.

Until recently I largely forgot about this, although I still thought of computers as really stupid because AutoCorrect was not and could not be made case-sensitive as I always assumed it should be. However, a discussion with my brother confirmed what he had said to be a quarter of a century ago — that AutoCorrect exists purely to correct typos, and is not designed to correct swearing (my brother’s “AutoCensor”). Although it ought to be simply enough to have separate AutoCorrect entries with different capitalisations, that has never been done because it would be more complex and the purpose was and is always corrections whose necessity is independent of capitalisation. If that be recognised, then computers that correct “Dick” when capitalised are simply doing what they are ask, whether it was my intention or not, and are not totally stupid as I have always thought!