Comparing the prime-rich centuries

I have always had an extremely strong interest in prime numbers — most likely because of the sense of order prime and prime factor tables provide.

From the time I saw the “First One Hundred Prime Numbers” in my old Mathematics Around Us: Skills and Applications book I had in school in the 1980s, I have always been interested in memorising prime tables as high as I have. In practice, I have not memorised my prime numbers and factors above the second millennium, although I do remember many primes in later millennia.

Over the past couple of years, I have discovered that the apparent lack of extremely prime-rich centuries above the century from 2,704,900 to 2,704,999 (which contains seventeen primes) actually does not apply for extremely large numbers. No eight-digit century contains more than fifteen primes and only six contain that many, but so early as 1904, American mathematician Leonard Eugene Dickson predicted that there must exist larger centuries with more than fifteen primes. After 2,704,900 there are none until the sixteen-prime century from 839,296,300 to 839,296,399 — 2.49175625272 orders of magnitude larger in base 10 and 5.73748080288 orders larger in base e. Evidence for Dickson’s Conjecture can however be seen in there being 26 nine-digit centuries, 37 ten-digit centuries, and over 100 eleven-digit centuries containing fifteen primes.

Another notable discovery is the existence of centuries with eighteen primes — predicted by Dickson in 1904 but not discovered until much later (I have no knowledge of when the first eighteen-prime century was discovered). Early writings about primes in centuries do not note Dickson’s work, but equally do not say that larger centuries with more primes than then-known centuries after the first two cannot exist.

The high density of primes in these recently discovered prime-rich centuries has made me want to compile them to remember the patterns of primes. Doing so is not easy in a prime-rich sequence of large numbers because there are so many unpredictable patterns of digits to remember — although some of the relevant centuries, like the fifth seventeen-prime century’s k of 144,997,771, do have a pattern that does aid memory a little. So I have decided to tabulate the first eleven seventeen- and eighteen-prime centuries. The second century in each sequence, however, has been excluded because k (the multiple of 100 beginning the century under examination) is not of the form 3n+1 as all the other first eleven seventeen- and eighteen-prime centuries are, but of the form 3n+2. Centuries where k is of the form 3n+2 (or 3n) contain two fewer numbers not divisible by 2, 3, or 5 than centuries where k is of the firm 3n+1 — hence they would require more numbers to properly tabulate and are much harder to compare with other centuries being examined.

First Eleven Centuries (excluding 1,400 to 1,499) with Seventeen Primes

4	7837	27049	144997771	651186838	12779564974	22369949923	149621468452	225012717952	240728320642
401	783,701	2,704,901			1,277,956,497,401	2,236,994,992,301	14,962,146,845,201	22,501,271,795,201
	783,703	2,704,903		65,118,683,803		2,236,994,992,303
	783,707	2,704,907	14,499,777,107	65,118,683,807	1,277,956,497,407	2,236,994,992,307	14,962,146,845,207		24,072,832,064,207
409		2,704,909	14,499,777,109	65,118,683,809	1,277,956,497,409		14,962,146,845,209	22,501,271,795,209	24,072,832,064,209
			14,499,777,113	65,118,683,813	1,277,956,497,413	2,236,994,992,313	14,962,146,845,213	22,501,271,795,213	24,072,832,064,213
419	783,719		14,499,777,119			2,236,994,992,319			24,072,832,064,219
421	783,721		14,499,777,121			2,236,994,992,321		22,501,271,795,221	24,072,832,064,221
		2,704,927			1,277,956,497,427	2,236,994,992,327	14,962,146,845,227	22,501,271,795,227	24,072,832,064,227
431		2,704,931			1,277,956,497,431			22,501,271,795,231	24,072,832,064,231
433	783,733		14,499,777,133						24,072,832,064,233
	783,737	2,704,937	14,499,777,137	65,118,683,837	1,277,956,497,437		14,962,146,845,237	22,501,271,795,237
439		2,704,939			1,277,956,497,439	2,236,994,992,339
443	783,743	2,704,943	14,499,777,143	65,118,683,843				22,501,271,795,243	24,072,832,064,243
449	783,749			65,118,683,849	1,277,956,497,449		14,962,146,845,249
			14,499,777,151		1,277,956,497,451		14,962,146,845,251		24,072,832,064,251
457		2,704,957		65,118,683,857	1,277,956,497,457	2,236,994,992,357	14,962,146,845,257	22,501,271,795,257
461			14,499,777,161			2,236,994,992,361
463	783,763	2,704,963	14,499,777,163	65,118,683,863	1,277,956,497,463	2,236,994,992,363
467	783,767		14,499,777,167	65,118,683,867		2,236,994,992,367	14,962,146,845,267
		2,704,969	14,499,777,169			2,236,994,992,369		22,501,271,795,269
					1,277,956,497,473	2,236,994,992,373	14,962,146,845,273	22,501,271,795,273	24,072,832,064,273
479	783,779	2,704,979	14,499,777,179	65,118,683,879	1,277,956,497,479		14,962,146,845,279	22,501,271,795,279	24,072,832,064,279
	783,781	2,704,981	14,499,777,181	65,118,683,881	1,277,956,497,481	2,236,994,992,381	14,962,146,845,281	22,501,271,795,281
487	783,787	2,704,987		65,118,683,887	1,277,956,497,487		14,962,146,845,287	22,501,271,795,287	24,072,832,064,287
491	783,791		14,499,777,191	65,118,683,891		2,236,994,992,391	14,962,146,845,291	22,501,271,795,291	24,072,832,064,291
	783,793	2,704,993		65,118,683,893			14,962,146,845,293	22,501,271,795,293	24,072,832,064,293
		2,704,997	14,499,777,197	65,118,683,897	1,277,956,497,497			22,501,271,795,297	24,072,832,064,297
499	783,799			65,118,683,899		2,236,994,992,399	14,962,146,845,299		24,072,832,064,299

One might note that there is a fairly random list of numbers being prime in these ten centuries. As one might expect, a higher proportion of the larger centuries (excluding those beginning with 400, 783,700 and 2,704,900) consists of very large constellations of primes. I noticed this very clearly in the century beginning with 65,118,683,800 (the sixth century containing seventeen primes) that most of the primes were two very large constellations including a prime quadruple, and this is more marked for some of the bigger centuries.

First Eleven Centuries (excluding 2,335,286,971,401,800 to 2,335,286,971,401,899) with Eighteen Primes

1228537713709	28703737474266	144785865481702	161394923966449	168975708209638	174748809066898	207552241231357	278215179205531	312303328909720	592248982143877
122,853,771,370,901	2,870,373,747,426,601		16,139,492,396,644,901	16,897,570,820,963,801			27,821,517,920,553,101	31,230,332,890,972,001	59,224,898,214,387,701
122,853,771,370,903		14,478,586,548,170,203		16,897,570,820,963,803	17,474,880,906,689,803	20,755,224,123,135,703	27,821,517,920,553,103	31,230,332,890,972,003
122,853,771,370,907	2,870,373,747,426,607		16,139,492,396,644,907			20,755,224,123,135,707	27,821,517,920,553,107		59,224,898,214,387,707
	2,870,373,747,426,609	14,478,586,548,170,209	16,139,492,396,644,909	16,897,570,820,963,809	17,474,880,906,689,809	20,755,224,123,135,709	27,821,517,920,553,109	31,230,332,890,972,009	59,224,898,214,387,709
	2,870,373,747,426,613		16,139,492,396,644,913			20,755,224,123,135,713		31,230,332,890,972,013	59,224,898,214,387,713
122,853,771,370,919		14,478,586,548,170,219	16,139,492,396,644,919		17,474,880,906,689,819
122,853,771,370,921	2,870,373,747,426,621	14,478,586,548,170,221	16,139,492,396,644,921	16,897,570,820,963,821	17,474,880,906,689,821	20,755,224,123,135,721	27,821,517,920,553,121	31,230,332,890,972,021	59,224,898,214,387,721
122,853,771,370,927	2,870,373,747,426,627			16,897,570,820,963,827	17,474,880,906,689,827	20,755,224,123,135,727	27,821,517,920,553,127	31,230,332,890,972,027
122,853,771,370,931		14,478,586,548,170,231	16,139,492,396,644,931		17,474,880,906,689,831		27,821,517,920,553,131	31,230,332,890,972,031
122,853,771,370,933	2,870,373,747,426,633		16,139,492,396,644,933
122,853,771,370,937		14,478,586,548,170,237	16,139,492,396,644,937		17,474,880,906,689,837	20,755,224,123,135,737
				16,897,570,820,963,839	17,474,880,906,689,839	20,755,224,123,135,739			59,224,898,214,387,739
	2,870,373,747,426,643	14,478,586,548,170,243	16,139,492,396,644,943	16,897,570,820,963,843			27,821,517,920,553,143	31,230,332,890,972,043	59,224,898,214,387,743
122,853,771,370,949		14,478,586,548,170,249		16,897,570,820,963,849	17,474,880,906,689,849				59,224,898,214,387,749
122,853,771,370,951	2,870,373,747,426,651		16,139,492,396,644,951		17,474,880,906,689,851	20,755,224,123,135,751	27,821,517,920,553,151	31,230,332,890,972,051
	2,870,373,747,426,657	14,478,586,548,170,257		16,897,570,820,963,857			27,821,517,920,553,157	31,230,332,890,972,057	59,224,898,214,387,757
122,853,771,370,961	2,870,373,747,426,661		16,139,492,396,644,961		17,474,880,906,689,861				59,224,898,214,387,761
		14,478,586,548,170,263	16,139,492,396,644,963	16,897,570,820,963,863		20,755,224,123,135,763	27,821,517,920,553,163	31,230,332,890,972,063
		14,478,586,548,170,267		16,897,570,820,963,867	17,474,880,906,689,867	20,755,224,123,135,767	27,821,517,920,553,167		59,224,898,214,387,767
122,853,771,370,969	2,870,373,747,426,669					20,755,224,123,135,769	27,821,517,920,553,169	31,230,332,890,972,069	59,224,898,214,387,769
122,853,771,370,973	2,870,373,747,426,673	14,478,586,548,170,273	16,139,492,396,644,973	16,897,570,820,963,873	17,474,880,906,689,873	20,755,224,123,135,773	27,821,517,920,553,173	31,230,332,890,972,073
	2,870,373,747,426,679	14,478,586,548,170,279	16,139,492,396,644,979	16,897,570,820,963,879	17,474,880,906,689,879	20,755,224,123,135,779		31,230,332,890,972,079	59,224,898,214,387,779
		14,478,586,548,170,281		16,897,570,820,963,881		20,755,224,123,135,781			59,224,898,214,387,781
122,853,771,370,987	2,870,373,747,426,687	14,478,586,548,170,287		16,897,570,820,963,887	17,474,880,906,689,887	20,755,224,123,135,787	27,821,517,920,553,187	31,230,332,890,972,087
122,853,771,370,991	2,870,373,747,426,691	14,478,586,548,170,291		16,897,570,820,963,891	17,474,880,906,689,891	20,755,224,123,135,791	27,821,517,920,553,191	31,230,332,890,972,091	59,224,898,214,387,791
	2,870,373,747,426,693	14,478,586,548,170,293	16,139,492,396,644,993	16,897,570,820,963,893	17,474,880,906,689,893		27,821,517,920,553,193		59,224,898,214,387,793
122,853,771,370,997			16,139,492,396,644,997	16,897,570,820,963,897	17,474,880,906,689,897	20,755,224,123,135,797	27,821,517,920,553,197	31,230,332,890,972,097	59,224,898,214,387,797
122,853,771,370,999	2,870,373,747,426,699	14,478,586,548,170,299	16,139,492,396,644,999					31,230,332,890,972,099	59,224,898,214,387,799

This table — more difficult to write up than for the seventeen-prime centuries because Excel cannot calculate large enough numbers — is rather odd because there seems to be a pattern of presence and absence in the first eighteen-prime centuries:

• k21 is always present as a prime
• k73 is present in the first nine
• k91 is only missing from one of these centuries
• contrariwise, k19, k33, k39 and k81 are missing from a majority of these centuries

What these results suggest I am not sure, but there may be some restrictions on prime patterns for centuries with eighteen primes that are not calculated in conventional theories. However, I have no evidence in tables or discussions that this is the case, so the variations may be mere coincidence.