How unusual were US winters in the Carter and early Reagan eras? An analysis using extended winter temperatures

The late 1984 article ‘Recent Unusual Mean Winter Temperatures Across the Contiguous United States’ by Thomas R. Karl, Robert E. Livezey and Edward S. Epstein showed a sequence of extremely unusual winters occurred across the United States beginning with the winter of 1975/1976 and continuing up to the then-current winter of 1983/1984.

Whilst the article is interesting, the knowledge of climate history I have gathered always appears to say that the period of the Great Depression from 1928/1929 to 1936/1937 saw winters as unusual —both in warmth and cold — as the period under review. An earlier analysis I did a year or two ago did not suggest this. This applies even when the two periods are analysed region by region. This last step is very important since I have long noted that many Depression winters saw extreme but opposite sign temperature anomalies in the West and East (and also in Alaska, which under stable global temperatures forms a three-way seesaw with the two contiguous regions). I also noticed studying the article that there was a remarkable absence of extreme winters in the contiguous United States between 1954/1955 and 1974/1975. During these twenty-one years, not one winter was outside 1.179 standard deviations from the mean — an occurrence with probability under 0.2 percent. This is noticed briefly also in ‘Agriculture and the Recent “Benign Climate” in Minnesota’ by Donald G. Baker, David L. Ruschy and Richard H. Skaggs.

As an extension of ‘Recent Unusual Mean Winter Temperatures Across the Contiguous United States’, I have attempted to do an analysis of CONUS temperatures over the extended winter seasons from November to March and from October to March, using the same criteria found in the original study. Karl, Livezey and Epstein showed that a departure of 1.253 standard deviations from the CONUS-averaged mean qualifies a season as “abnormal” in that the proportion of a normal distribution outside such a range is less than one-tenth the complete set of data. However, they also noted a range of 1.179 standard deviations from the mean to give a slightly less rigid definition of an “unusual” season. With this noted, I will analyse the extended winter seasons to see if they produced the same abnormalities Karl, Livezey and Epstein noticed for the December to February season.

November to March:

Mean area-averaged contiguous US temperature from November to March between 1895/1896 and 1983/1984, with mean shown with dashed line and 1.179 standard deviations above and below mean with solid line

Year	Contiguous U.S. Average T, November to March (˚C)	Standardised anomaly
1895/1896	1.71	-0.46
1896/1897	1.58	-0.64
1897/1898	2.08	+0.06
1898/1899	-0.02	-2.84
1899/1900	2.52	+0.66
1900/1901	2.02	-0.03
1901/1902	1.81	-0.33
1902/1903	1.77	-0.37
1903/1904	1.29	-1.03
1904/1905	1.59	-0.63
1905/1906	1.77	-0.38
1906/1907	3.03	+1.36
1907/1908	2.76	+0.99
1908/1909	2.61	+0.78
1909/1910	2.11	+0.09
1910/1911	2.63	+0.81
1911/1912	0.12	-2.66
1912/1913	1.53	-0.70
1913/1914	2.58	+0.74
1914/1915	1.28	-1.05
1915/1916	2.18	+0.19
1916/1917	0.65	-1.92
1917/1918	1.84	-0.27
1918/1919	2.21	+0.23
1919/1920	1.26	-1.09
1920/1921	3.41	+1.89
1921/1922	1.87	-0.24
1922/1923	1.98	-0.08
1923/1924	1.91	-0.18
1924/1925	2.40	+0.49
1925/1926	2.28	+0.33
1926/1927	2.61	+0.78
1927/1928	2.44	+0.55
1928/1929	1.26	-1.09
1929/1930	1.82	-0.30
1930/1931	2.51	+0.65
1931/1932	2.67	+0.87
1932/1933	1.77	-0.38
1933/1934	3.46	+1.95
1934/1935	3.20	+1.60
1935/1936	0.69	-1.86
1936/1937	1.08	-1.33
1937/1938	2.96	+1.27
1938/1939	2.28	+0.33
1939/1940	2.13	+0.13
1940/1941	2.22	+0.24
1941/1942	2.32	+0.39
1942/1943	2.26	+0.29
1943/1944	2.06	+0.02
1944/1945	2.61	+0.78
1945/1946	2.68	+0.88
1946/1947	2.21	+0.23
1947/1948	1.03	-1.40
1948/1949	1.42	-0.86
1949/1950	2.59	+0.76
1950/1951	1.88	-0.23
1951/1952	1.61	-0.60
1952/1953	3.27	+1.69
1953/1954	3.24	+1.65
1954/1955	2.05	+0.01
1955/1956	1.55	-0.68
1956/1957	2.55	+0.70
1957/1958	2.06	+0.02
1958/1959	2.07	+0.04
1959/1960	1.17	-1.21
1960/1961	2.63	+0.82
1961/1962	1.26	-1.08
1962/1963	2.09	+0.07
1963/1964	1.56	-0.66
1964/1965	1.57	-0.65
1965/1966	2.34	+0.42
1966/1967	2.83	+1.09
1967/1968	2.12	+0.10
1968/1969	1.04	-1.38
1969/1970	1.78	-0.36
1970/1971	1.94	-0.14
1971/1972	2.57	+0.72
1972/1973	1.81	-0.33
1973/1974	2.97	+1.28
1974/1975	2.21	+0.23
1975/1976	3.19	+1.58
1976/1977	1.30	-1.02
1977/1978	1.25	-1.09
1978/1979	0.38	-2.30
1979/1980	2.14	+0.14
1980/1981	3.26	+1.68
1981/1982	2.13	+0.13
1982/1983	3.26	+1.67
1983/1984	1.64	-0.55

October to March:

Mean area-averaged contiguous US temperature from October to March between 1895/1896 and 1983/1984, with mean shown with dashed line and 1.179 standard deviations above and below mean with solid line

Year	Contiguous U.S. Average T, October to March (˚C)	Standardised anomaly
1895/1896	3.17	-0.87
1896/1897	3.16	-0.89
1897/1898	3.87	+0.19
1898/1899	1.79	-2.98
1899/1900	4.13	+0.59
1900/1901	3.94	+0.30
1901/1902	3.67	-0.11
1902/1903	3.56	-0.29
1903/1904	3.11	-0.97
1904/1905	3.36	-0.59
1905/1906	3.29	-0.69
1906/1907	4.43	+1.04
1907/1908	4.32	+0.87
1908/1909	4.01	+0.40
1909/1910	3.69	-0.08
1910/1911	4.37	+0.96
1911/1912	2.01	-2.64
1912/1913	3.24	-0.77
1913/1914	3.97	+0.34
1914/1915	3.22	-0.81
1915/1916	3.98	+0.36
1916/1917	2.41	-2.04
1917/1918	3.24	-0.77
1918/1919	4.08	+0.52
1919/1920	2.95	-1.21
1920/1921	4.94	+1.82
1921/1922	3.72	-0.04
1922/1923	3.78	+0.06
1923/1924	3.37	-0.57
1924/1925	4.13	+0.58
1925/1926	3.47	-0.42
1926/1927	4.30	+0.85
1927/1928	4.25	+0.77
1928/1929	3.18	-0.86
1929/1930	3.55	-0.30
1930/1931	3.92	+0.26
1931/1932	4.50	+1.15
1932/1933	3.35	-0.60
1933/1934	5.03	+1.96
1934/1935	4.94	+1.82
1935/1936	2.59	-1.75
1936/1937	2.89	-1.31
1937/1938	4.49	+1.14
1938/1939	4.16	+0.64
1939/1940	3.88	+0.20
1940/1941	4.08	+0.51
1941/1942	4.08	+0.52
1942/1943	3.97	+0.34
1943/1944	3.72	-0.03
1944/1945	4.33	+0.89
1945/1946	4.28	+0.81
1946/1947	3.80	+0.08
1947/1948	3.34	-0.62
1948/1949	3.14	-0.92
1949/1950	4.20	+0.69
1950/1951	3.92	+0.27
1951/1952	3.37	-0.57
1952/1953	4.66	+1.40
1953/1954	4.94	+1.82
1954/1955	3.81	+0.09
1955/1956	3.42	-0.49
1956/1957	4.36	+0.94
1957/1958	3.53	-0.32
1958/1959	3.83	+0.14
1959/1960	2.94	-1.22
1960/1961	4.30	+0.85
1961/1962	3.07	-1.03
1962/1963	3.98	+0.36
1963/1964	3.83	+0.14
1964/1965	3.33	-0.64
1965/1966	4.08	+0.52
1966/1967	4.29	+0.83
1967/1968	3.79	+0.07
1968/1969	2.97	-1.19
1969/1970	3.25	-0.75
1970/1971	3.49	-0.39
1971/1972	4.28	+0.81
1972/1973	3.38	-0.55
1973/1974	4.71	+1.47
1974/1975	3.86	+0.18
1975/1976	4.75	+1.53
1976/1977	2.74	-1.53
1977/1978	3.08	-1.02
1978/1979	2.39	-2.07
1979/1980	3.94	+0.30
1980/1981	4.62	+1.34
1981/1982	3.66	-0.13
1982/1983	4.68	+1.43
1983/1984	3.50	-0.37

Analysis and Conclusions:

The tables above suggest that, when we extend the winter season to November to March, and to October to March, we find that the period from 1975/1976 to 1983/1984 was not so unusual as Karl, Livezey and Epstein suggested. Only four of the nine November to March periods were outside the range described in their article, and only five of the nine October to March periods.

We also find that the period from 1954/1955 to 1974/1975 was not so exceptional for lacking extremes when we extend the winter season. Based on a November to March or an October to March period, 1959/1960 and 1968/1969 were “unusually” cold, and 1973/1974 unusually warm, though 1959/1960 is a marginal case on both periods, and October 1968 to March 1969 is likewise. 1973/1974, though, is especially warm in the Deep South, where January 1974 remains the warmest on record in South Carolina, Georgia and Florida.

Standardized contiguous US temperature anomalies relative to 1895 to 2000 mean for October 1959 to March 1960. Note area in southeastern Nebraska and northern Kansas over two standard deviations below the mean, and extensive area in West South Central and Plains colder than 1.5 standard deviations below the mean

Standardized contiguous US temperature anomalies relative to 1895 to 2000 mean for October 1973 to March 1974. Note wide area of the South hotter than 1.5 standard deviations above average.

Standardized contiguous US temperature anomalies relative to 1895 to 2000 mean for November 1968 to March 1969. Note widespread area colder than 1.5 standard deviations below mean from Florida to Montana

We can also note that the winter of 1981/1982 is slightly above normal when measured on the basis of November to March, and only slightly below on an October to March basis. This is because November 1981 was from coast to coast the hottest since 1949 and at the time the sixth-hottest since 1895. Over Minnesota it was the warmest since 1899, over North Dakota the warmest since 1949, and over South Dakota and Nebraska the warmest since 1954, although Monthly Weather Review failed to hint at this via reference to long-term stations there.

November CONUS temperature from 1895 to 1997, with 1895 to 1974 mean of 41.54˚F or 5.3˚C. Note November 1981’s warmth, previously exceeded only by Novembers 1949, 1934, 1913, 1909 and 1899

The very long Fort Snelling, Minnesota record — the oldest temperature station west of the Appalachian Mountains — shows November 1830 and November 1849 as over 1˚C warmer than any November during the twentieth century, not being exceeded until November 2001 when Australian coal barons and Gulf oil sheikhs had taken firm control of the climate. Nonetheless, November 1981 was relative to the preindustrial mean more abnormally warm over the contiguous US overall than January 1982 was cold, so it makes sense that the extended 1981/1982 winter was not cold.

Standardised mean temperature anomalies by climate division over the contiguous US for November 1981. Note the anomalous warmth in the Great Plains and Minnesota (where it was the warmest for 82 years)

The very cold winter of 1977/1978 also appears less abnormal when measured over the extended winter, again largely due to a relatively warm November (though not nearly so warm as November 1981). 1976/1977 is affected by a very warm March, which would lead into the hottest spring/summer season from coast to coast since 1936 (ironically, also the last colder winter!). The winter of 1947/1948, on a November to March basis, also stands out as abnormally cold. [1947/1948 is not exceptional on an October to March basis since October 1947 was exceptionally hot — the hottest by mean minimum temperature of the century, and the second-hottest by mean temperature.]

Standardised divisional temperature anomalies for the period from November 1947 to March 1948. Note the general cold outside of Florida, and the area in the Southwest colder than two standard deviations below the mean

The Depression Era:

An earlier cursory examination, and basic knowledge of contiguous US temperature maps, has made me suggest that the winters discussed by Karl, Livezey and Epstein for the Carter and early Reagan eras were replicated during the Great Depression. The table on page 1304 notes several abnormal winters in the Depression era in 1930/1931 (marginal), 1931/1932, 1933/1934 and 1935/1936. However, at a local divisional level, it is clear that almost all the winters from 1928/1929 to 1936/1937 were highly abnormal. The exceptions I noted were 1929/1930 and 1934/1935, but the former is practically disqualified, for it saw a record cold January followed by a record warm February.

The tables above, however, do show that the extended winters during the 1930s were in fact more abnormal than those of the late 1970s and early 1980s. Five consecutive years from 1933/1934 to 1937/1938 qualify as abnormal based on November to March temperature, and the first four are also abnormal based upon October to March temperature. 1934/1935 is particularly striking despite the winter’s apparent absence of large regional anomalies: in fact it is the equal second-warmest October to March between at least 1883 and 1991:

Standardised divisional anomalies for October 1934 to March 1935. Note departures exceeding plus two standard deviations in Southern Plains, and 2.5 standard deviations in division OK 1

In fact, in climate divisions OK 1 [Cimarron, Texas, Beaver, Harper and Ellis Counties], and KS 7 [Southwest], this was the warmest October to March period until 2016:

For Oklahoma as a whole, 1934/1935 was the warmest October to March until 2000. The extended winter of 1936/1937 featured not merely a record cold January west of the Mississippi, but also general cold in October, November and March. 1937/1938, although only marginal for the list above, and not qualifying by an October to March period at all, was warmer than normal virtually everywhere and substantially so in the South and the Intermountain West:

Divisional monthly mean temperature standardised anomalies relative to 1895 to 2000 mean for November 1937 to March 1938. Note striking absence of cooler than average areas

Thus, this study shows my suspicion that Great Depression-era winters were just as abnormal as those of the late 1970s and early 1980s to be correct. This becomes emphatically true when one observes the extreme regional divides found in several of the early 1930s, and the extreme contrast between then-record cold January 1930 and then-record warm February 1930. All these contrasts were noted eleven years ago here.

Beyond Climate Division Era — A Possible Analogue in the Post-Reconstruction Years:

Some time ago, looking to expand knowledge of US climate history beyond the GISS temperature maps (which go back to 1880), I noticed a possible parallel — at least over the Midwest — between approximately 1875 and 1890. Studying the data for Dubuque, Iowa — where temperature data go back to 1851, or twenty years further than most of the US outside the Northeast — I noticed a large number of abnormal winters during those post-Reconstruction years. I observed especially how six of the twelve warmest winters in Dubuque between 1851 and 1975 occurred in this era:

Mean winter temperature in Dubuque, Iowa from 1851/1852 to 1983/1984 [later year labelled on x-axis], with -1.179σ and +1.179σ marked by solid lines, and all-series mean to 1984 by dashed line.

The graph does show a very clear concentration of abnormal winters between 1875 and 1890. Although a comparison with Fort Snelling, Minnesota suggests the Dubuque shelters in use at this time may have given hotter temperatures than a standard modern shelter, this does not refute the argument for an unusual concentration of abnormal Midwestern winters in the late 1870s and 1880s, and the stark contrast with the 1860s. If we extend the winter seasons for Dubuque as we did in the first experiment in this post, we find that the abnormality of post-Reconstruction winters largely remains, although the warm seasons from 1888/1889 to 1891/1892 become fairly unexceptional:

Mean November to March temperature in Dubuque, Iowa from 1851/1852 to 1983/1984 [later year labelled on x-axis], with -1.179σ and +1.179σ marked by solid lines, and all-series mean to 1984 by dashed line.

Mean November to March temperature in Dubuque, Iowa from 1851/1852 to 1983/1984 [later year labelled on x-axis], with -1.179σ and +1.179σ marked by solid lines, and all-series mean to 1984 by dashed line.

Despite the unexceptional nature of the seasons from 1888/1889 to 1891/1892, there remains a sufficient number of abnormal winters in the 1870s and 1880s to make the “exceptional” period a century later seem much less so. It is unfortunate that for the western half of the United States too few temperature data exist to discover whether the contrast between an absence of extreme winters in the 1860s and an abundance thereof in the 1870s and 1880s was replicated there. Data from Boise, Idaho — the oldest station in the Northwest — suggest that:

1. the winters of the late 1870s and 1880s were less abnormal than in the Midwest, and most are unlikely to be abnormal except when the extended season is used
2. the winters of the 1930s and from the middle 1970s to the middle 1980s were exceptionally abnormal in this region of the United States

How socialist groups survive when manufacturing disappears — and why they cannot adapt

This afternoon, walking past the SLV in a wide-ranging, although not lengthy, shopping trip, I saw something that was both extremely familiar and at the same time shocking.

This was the exceedingly familiar pro-Palestine messages:

1. “FREE, FREE, FREE PALESTINE”
2. “FROM THE RIVER TO THE SEA, PALESTINE WILL BE FREE”

but written in chalk on the ledges at the western exit from the SLV.

The resemblance of the writing to graffiti — although I was extremely familiar with the use of graffiti by anarchist groups at various universities around Melbourne — genuinely surprised me. I assumed that pro-Palestine groups would always use posters as a means of communication.

The unfortunate consequence was that I became obsessed with the familiar slogans at once, and could not do anything except switch between Trotskyist dogmatism and its antithesis in books like Robert Spencer’s The Palestinian Delusion: The Catastrophic History of the Middle Eastern Peace Process. Both groups seemed to show the common mainstream arguments as completely wrong, and I have not read a rigorous refutation of the most controversial arguments of either by mainstream analysts.

Examples are the refusal to accept Jewish refugees and that

“Thus, by the 1920s, the Zionists had been marginalized on all sides. The majority of the world's Jews clearly showed their desire to emigrate to Western countries. And thousands of Jews who remained in Eastern Europe fought for a better life, winning solidarity from many of their Gentile brothers and sisters.”

as noted by Lance Selfa, the role of jihad in the conflict as noted by Spencer, and the “Jewish nakba” noted by Spencer but much more by Joan Peters.

However, a conversation today with my brother tonight about the graffiti said something that is really disturbing. This is that socialist organisations deprived by globalisation of what should be their financial and membership base — militant workers in labour-intensive manufacturing — have turned to anti-Zionist Muslim migrants. What my brother noted is that every single socialist parliamentary candidate joins pro-Palestine rallies wherever and whenever they are held. Yet, these rallies are tightly linked with Islamist groups who would outlaw any radically democratic secular party and execute its leaders. Worse still, they are ultimately linked with hyper-capitalist petromonarchies like Qaṭar, whom by all logic socialist groups.should.condemn much more strongly than Israel, given that the philosophy behind these pre-Enlightenment states is precisely the opposite of The Communist Manifesto. The worst thing is that this deep intetwining, which dates back at least to Lenin’s attempted alliance with Ibn Sa^cud, makes it.impossible for socialist groups to do anything to update a wordview minimally fifty-three years outdated.

Some notable fourteen-prime centuries

Today, I had a look for “full digits” in fourteen-and fifteen-prime centuries between the beginning of the “eight-digit gap” and 10¹¹ (100 billion or 100,000,000).

Given my finding that the 1,790,288^th century — overall the thirty-seventh century with exactly fifteen prime numbers — contained seven primes ending in 9, it was only natural that I would seek to search for other fifteen- and fourteen-prime centuries with seven primes ending in one digit. [The 80,563^rd and 768,053^rd centuries with thirteen primes I already knew as having seven primes ending in one digit, so that count did not need discussion. Smaller counts would be too unlikely to have so many primes ending in a single digit, although because centuries with fewer primes are so much more numerous there are actually more “full”-digit centuries with fewer than thirteen total primes].

Early checks of my sequences of centuries with seven primes ending in one digit versus OEIS sequence A186406 did not find a case where there were fourteen total primes and seven ending in one digit as far as I could search. I did another check as far as I could go using the method of modulo(3003) discussed here and checked to see if further centuries in A186406 had a potential “full” century. After I finished my work I thought that I ought to check the fifteen-prime centuries I within the same range of magnitudes.

First 14-Prime century with Empty Digit: “Empty-1” 1,015,316^th:

101531501 = 229 × 443369
1. 101531503 is prime [1.]
101531507 = 7 × 11 × 59 × 22349
1. 101531509 is prime [1.]
2. 101531513 is prime [2.]
3. 101531519 is prime [2.]
101531521 = 7 × 13 × 1115731
4. 101531527 is prime [1.]
101531531 = 17 × 5972443
5. 101531533 is prime [3.]
6. 101531537 is prime [2.]
101531539 = 41 × 73 × 33923
7. 101531543 is prime [4.]
101531549 = 7 × 97 × 149531
101531551 = 11 × 107 × 86263
101531557 = 43 × 2361199
101531561 = 9133 × 11117
101531563 = 7 × 101 × 143609
8. 101531567 is prime [3.]
9. 101531569 is prime [3.]
101531573 = 11 × 13 × 19 × 37369
10. 101531579 is prime [4.]
101531581 = 29 × 3501089
11. 101531587 is prime [4.]
101531591 = 7 × 23 × 199 × 3169
12. 101531593 is prime [5.]
13. 101531597 is prime [5.]
101531599 = 13 × 17 × 67 × 6857

This century is a little strange given that, as the first fourteen-prime century with no primes ending in one of 1, 3, 7, or 9, it is the 116^th century with fourteen primes — those easy to remember in a table because it is the smallest with nine rather than eight digits. The century from 252,724,900 to 252,724,999 with fifteen primes but none ending in 7, is the fortieth century with fifteen primes. The first thirteen-prime century with no primes ending in one of 1, 3, 7, or 9 is the century from 63,600 to 63,699 (no primes ending in 3; overall the 36^th century with thirteen primes), and the first such twelve-prime century is that from 16,400 to 16,499 (no primes ending in 9; overall the 39^th century with twelve primes).

The nine centuries beginning with 200 produce via deletion three clear fourteen-prime patterns with no primes ending in one digit:

1. {7, 13, 17, 37, 47, 49, 53, 59, 67, 73, 79, 83, 89, 97} via deletion from the fourth century has fourteen primes with none ending in 1
2. {1, 7, 13, 17, 31, 41, 43, 47, 53, 61, 73, 77, 83, 91} via deletion from the seventh century has fourteen primes with none ending in 9
3. {9, 13, 19, 21, 31, 33, 39, 49, 51, 61, 63, 69, 91, 93} via deletion from the eleventh century has fourteen primes with none ending in 7

Contrariwise, no fifteen-prime pattern with no primes ending in one digit can be created via deletion from any century between 200 and 252,724,899.

The Only “Full Digit” 14-Prime Century in First Thousand: “Full 7” 113,684,594^th:

1. 11368459301 is prime
11368459303 = 101 × 139 × 263 × 3079
2. 11368459307 is prime [1.]
11368459309 = 37 × 59 × 5207723
3. 11368459313 is prime
11368459319 = 7 × 811 × 2002547
4. 11368459321 is prime
5. 11368459327 is prime [2.]
11368459331 = 29 × 2243 × 174773
11368459333 = 7 × 11 × 157 × 487 × 1931
6. 11368459337 is prime [3.]
11368459339 = 2633 × 4317683
11368459343 = 23 × 233 × 2121377
11368459349 = 13 × 53 × 673 × 24517
11368459351 = 17 × 668732903
7. 11368459357 is prime [4.]
11368459361 = 7 × 13921 × 116663
8. 11368459363 is prime
9. 11368459367 is prime [5.]
11368459369 = 167 × 2333 × 29179
11368459373 = 19 × 598339967
10. 11368459379 is prime
11. 11368459381 is prime
12. 11368459387 is prime [6.]
11368459391 = 1523 × 7464517
13. 11368459393 is prime
14. 11368459397 is prime [7.]
11368459399 = 11 × 7879 × 131171

This is the only case amongst the first thousand fourteen-prime centuries where seven of the primes end in one digit. It is the 674^th century with exactly fourteen primes. It is surprising that this should be so when one sees that the “full 1” 80,563^rd century is only the 263^rd century with thirteen primes and the “full 3” 768,053^rd the 568^th. The surprise is even greater since a fourteen-prime century must logically have greater likelihood of a “full” digit than a thirteen-prime century: I recall that a thirteen-prime century had something like a 1-in-490 chance of a “full” digit and a fourteen-prime century closer to 1-in-340.

130,724,018^th century: 14 primes with gap of 36:

1. 13072401701 is prime
2. 13072401703 is prime
3. 13072401707 is prime
4. 13072401709 is prime
13072401713 = 7 × 12329 × 151471
13072401719 = 13 × 17 × 109 × 127 × 4273
5. 13072401721 is prime
13072401727 = 7 × 11 × 463 × 366677
6. 13072401731 is prime
7. 13072401733 is prime
8. 13072401737 is prime
13072401739 = 23 × 199 × 2856107
9.13072401743 is prime
13072401749 = 11 × 31 × 43 × 891523
10. 13072401751 is prime
11. 13072401757 is prime
12. 13072401761 is prime
13. 13072401763 is prime
13072401767 = 137 × 95418991
13072401769 = 7 × 9241 × 202087
13072401773 = 6379 × 2049287
13072401779 = 57787 × 226217
13072401781 = 73 × 179073997
13072401787 = 17 × 768964811
13072401791 = 89 × 331 × 443749
13072401793 = 11 × 19 × 577 × 108401
13072401797 = 7 × 13 × 143652767
14. 13072401799 is prime

Here we see a century with fourteen primes but a gap of thirty-six, which surprised me when I had a look, although it is less so when one studies the 240,728,320,643^rd century, where by deleting one could have fifteen primes with a gap of 40 in the middle, or much more “familiarly”, the amazing century from 15,640 to 15,740 with fourteen primes, two prime quadruples at the ends, and a record gap of 43 consecutive composite numbers between 15,683 and 15,727 in the middle.

The 219,562,919^th Century: A Prime Decaplet

21956291801 = 59 × 372140539
21956291803 = 13807 × 1590229
1. 21956291807 is prime
21956291809 = 17 × 23 × 56154199
2. 21956291813 is prime
21956291819 = 7 × 11 × 29 × 9832643
21956291821 = 53 × 414269657
3. 21956291827 is prime
21956291831 = 353 × 1693 × 36739
21956291833 = 7 × 19 × 19 × 41 × 73 × 2903
21956291837 = 71 × 397 × 778951
21956291839 = 349 × 62912011
21956291843 = 17 × 953 × 1355243
21956291849 = 31 × 6337 × 111767
21956291851 = 13 × 163 × 1831 × 5659
4. 21956291857 is prime
21956291861 = 7 × 7 × 448087589
21956291863 = 11 × 541 × 3689513
5. 21956291867 is prime
6. 21956291869 is prime
7. 21956291873 is prime
8. 21956291879 is prime
9. 21956291881 is prime
10. 21956291887 is prime
11. 21956291891 is prime
12. 21956291893 is prime
13. 21956291897 is prime
14. 21956291899 is prime

Here we see a fourteen-prime century ending with a prime decaplet — a group of ten primes in 33 integers, which is the shortest span in which ten consecutive primes can occur. OEIS sequences A027569 and A027570 show it as the third prime decaplet after those beginning with 11 and 9853497737. I had never seen anything like this, and checking a few other “small” decaplets from OEIS shows that most are in centuries with even fewer “extraneous” primes than the four in the century above.

The 819,751,072nd century — Eight Primes in 37 at End of Century:

81975107101 = 401 × 967 × 211403
1. 81975107107 is prime
2. 81975107111 is prime
3. 81975107113 is prime
81975107117 = 811 × 6053 × 16699
81975107119 = 83 × 509 × 1940377
81975107123 = 7 × 13 × 4637 × 194269
81975107129 = 73 × 461 × 2435893
81975107131 = 6947 × 11800073
81975107137 = 7 × 11 × 19 × 41 × 1366639
4. 81975107141 is prime
5. 81975107143 is prime
6. 81975107147 is prime
81975107149 = 13 × 39371 × 160163
81975107153 = 49871 × 1643743
81975107159 = 11 × 17 × 3767 × 116371
7. 81975107161 is prime
8. 81975107167 is prime
9. 81975107171 is prime
10. 81975107173 is prime
11. 81975107177 is prime
81975107179 = 7 × 7 × 107 × 15635153

12. 81975107183 is prime
13. 81975107189 is prime
14. 81975107191 is prime
15. 81975107197 is prime

This is a fifteen-prime century, but still quite interesting for the similarity to the previous section in the massive cluster at the end. What is actually more notable here is that although there are only eleven composite numbers coprime to 30, six of them for decades with no primes.

Pentatrigesimal multiplication table

As a response to seeing a relatively conventional hexadecimal multiplication table on Wikipedia, I am writing one for pentatrigesimal, as defined in the post here:

	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z	10
1	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F	G	H	I	J	K	L	M	N	P	Q	R	S	T	U	V	W	X	Y	Z	10
2	2	4	6	8	A	C	E	G	I	K	M	P	R	T	V	X	Z	11	13	15	17	19	1B	1D	1F	1H	1J	1L	1N	1Q	1S	1U	1W	1Y	20
3	3	6	9	C	F	I	L	P	S	V	Y	11	14	17	1A	1D	1G	1J	1M	1Q	1T	1W	1Z	22	25	28	2B	2E	2H	2K	2N	2R	2U	2X	30
4	4	8	C	G	K	P	T	X	11	15	19	1D	1H	1L	1Q	1U	1Y	22	26	2A	2E	2I	2M	2R	2V	2Z	33	37	3B	3F	3J	3N	3S	3W	40
5	5	A	F	K	Q	V	10	15	1A	1F	1K	1Q	1V	20	25	2A	2F	2K	2Q	2V	30	35	3A	3F	3K	3Q	3V	40	45	4A	4F	4K	4Q	4V	50
6	6	C	I	P	V	11	17	1D	1J	1Q	1W	22	28	2E	2K	2R	ZX	33	39	3F	3L	3S	3Y	44	4A	4G	4M	4T	4Z	55	5B	5H	5N	5U	60
7	7	E	L	T	10	17	1E	1L	1T	20	27	2E	2L	2T	30	37	3E	3L	3T	40	47	4E	4L	4T	50	57	5E	5L	5T	60	67	6E	6L	6T	70
8	8	G	P	X	15	1D	1L	1U	22	2A	2I	2R	2Z	37	3F	3N	3W	44	4C	4K	4T	51	59	5H	5Q	5Y	66	6E	6M	6V	73	7B	7J	7S	80
9	9	I	S	11	1A	1J	1T	22	2B	2K	2U	33	3C	3L	3V	44	4D	4M	4W	55	5E	5N	5X	66	6F	6P	6Y	77	7G	7Q	7Z	88	8H	8R	90
A	A	K	V	15	1F	1Q	20	2A	2K	2V	35	3F	3Q	40	4A	4K	4V	55	5F	5Q	60	6A	6K	6V	75	7F	7Q	80	8A	8K	8V	95	9F	9Q	A0
B	B	M	Y	19	1K	1W	27	2I	2U	35	3G	3S	43	4E	4Q	51	5C	5N	5Z	6A	6L	6X	78	7J	7V	86	8H	8T	94	9F	9R	A2	AD	AP	B0
C	C	P	11	1D	1Q	22	2E	2R	33	3F	3S	44	4G	4T	55	5H	5U	66	6I	6V	77	7J	7W	88	8K	8X	99	9L	9Y	AA	AM	AZ	BB	BN	C0
D	D	R	14	1H	1V	28	2L	2Z	3C	3Q	43	4G	4U	57	5K	5Y	6B	6P	72	7F	7T	86	8J	8X	9A	9N	A1	AE	AS	B5	BI	BW	C9	CM	D0
E	E	T	17	1L	20	2E	2T	37	3L	40	4E	4T	57	5L	60	6E	6T	77	7L	80	8E	8T	97	9L	A0	AE	AT	B7	BL	C0	CE	CT	D7	DL	E0
F	F	V	1A	1Q	25	2K	30	3F	3V	4A	4Q	55	5K	60	6F	6V	7A	7Q	85	8K	90	9F	9V	AA	AQ	B5	BK	C0	CF	CV	DA	DQ	E5	EK	F0
G	G	X	1D	1U	2A	2R	37	3N	44	4K	51	5H	5Y	6E	6V	7B	7S	88	8P	95	9L	A2	AI	AZ	BF	BW	CC	CT	D9	DQ	E6	EM	F3	FJ	G0
H	H	Z	1G	1Y	2F	ZX	3E	3W	4D	4V	5C	5U	6B	6T	7A	7S	89	8R	98	9Q	A7	AP	B6	BN	C5	CM	D4	DL	E3	EK	F2	FJ	G1	GI	H0
I	I	11	1J	22	2K	33	3L	44	4M	55	5N	66	6P	77	7Q	88	8R	99	9S	AA	AT	BB	BU	CC	CV	DD	DW	EE	EX	FF	FY	GG	GZ	HH	I0
J	J	13	1M	26	2Q	39	3T	4C	4W	5F	5Z	6I	72	7L	85	8P	98	9S	AB	AV	BE	BY	CH	D1	DK	E4	EN	F7	FR	GA	GU	HD	HX	IG	J0
K	K	15	1Q	2A	2V	3F	40	4K	55	5Q	6A	6V	7F	80	8K	95	9Q	AA	AV	BF	C0	CK	D5	DQ	EA	EV	FF	G0	GK	H5	HQ	IA	IV	JF	K0
L	L	17	1T	2E	30	3L	47	4T	5E	60	6L	77	7T	8E	90	9L	A7	AT	BE	C0	CL	D7	DT	EE	F0	FL	G7	GT	HE	I0	IL	J7	JT	KE	L0
M	M	19	1W	2I	35	3S	4E	51	5N	6A	6X	7J	86	8T	9F	A2	AP	BB	BY	CK	D7	DU	EG	F3	FQ	GC	GZ	HL	I8	IV	JH	K4	KR	LD	M0
N	N	1B	1Z	2M	3A	3Y	4L	59	5X	6K	78	7W	8J	97	9V	AI	B6	BU	CH	D5	DT	EG	F4	FS	GF	H3	HR	IE	J2	JQ	KD	L1	LP	MC	N0
P	P	1D	22	2R	3F	44	4T	5H	66	6V	7J	88	8X	9L	AA	AZ	BN	CC	D1	DQ	EE	F3	FS	GG	H5	HU	II	J7	JW	KK	L9	LY	MM	NB	P0
Q	Q	1F	25	2V	3K	4A	50	5Q	6F	75	7V	8K	9A	A0	AQ	BF	C5	CV	DK	EA	F0	FQ	GF	H5	HV	IK	JA	K0	KQ	LF	M5	MV	NK	PA	Q0
R	R	1H	28	2Z	3Q	4G	57	5Y	6P	7F	86	8X	9N	AE	B5	BW	CM	DD	E4	EV	FL	GC	H3	HU	IK	JB	K2	KT	LJ	MA	N1	NS	PI	Q9	R0
S	S	1J	2B	33	3V	4M	5E	66	6Y	7Q	8H	99	A1	AT	BK	CC	D4	DW	EN	FF	G7	GZ	HR	II	JA	K2	KU	LL	MD	N5	NX	PP	QG	R8	S0
T	T	1L	2E	37	40	4T	5L	6E	77	80	8T	9L	AE	B7	C0	CT	DL	EE	F7	G0	GT	HL	IE	J7	K0	KT	LL	ME	N7	P0	PT	QL	RE	S7	T0
U	U	1N	2H	3B	45	4Z	5T	6M	7G	8A	94	9Y	AS	BL	CF	D9	E3	EX	FR	GK	HE	I8	J2	JW	KQ	LJ	MD	N7	P1	PV	QP	RI	SC	T6	U0
V	V	1Q	2K	3F	4A	55	60	6V	7Q	8K	9F	AA	B5	C0	CV	DQ	EK	FF	GA	H5	I0	IV	JQ	KK	LF	MA	N5	P0	PV	QQ	RK	SF	TA	U5	V0
W	W	1S	2N	3J	4F	5B	67	73	7Z	8V	9R	AM	BI	CE	DA	E6	F2	FY	GU	HQ	IL	JH	KD	L9	M5	N1	NX	PT	QP	RK	SG	TC	U8	V4	W0
X	X	1U	2R	3N	4K	5H	6E	7B	88	95	A2	AZ	BW	CT	DQ	EM	FJ	GG	HD	IA	J7	K4	L1	LY	MV	NS	PP	QL	RI	SF	TC	U9	V6	W3	X0
Y	Y	1W	2U	3S	4Q	5N	6L	7J	8H	9F	AD	BB	C9	D7	E5	F3	G1	GZ	HX	IV	JT	KR	LP	MM	NK	PI	QG	RE	SC	TA	U8	V6	W4	X2	Y0
Z	Z	1Y	2X	3W	4V	5U	6T	7S	8R	9Q	AP	BN	CM	DL	EK	FJ	GI	HH	IG	JF	KE	LD	MC	NB	PA	Q9	R8	S7	T6	U5	V4	W3	X2	Y1	Z0
10	10	20	30	40	50	60	70	80	90	A0	B0	C0	D0	E0	F0	G0	H0	I0	J0	K0	L0	M0	N0	P0	Q0	R0	S0	T0	U0	V0	W0	X0	Y0	Z0	100