The 473,268th and 473,269th centuries — how few primes do you need?

In recent days, as I recover from a series of “bugs” (viruses) that badly affected me all through June and made me feel much chillier than usual in typically cool Melbourne winter weather, I have looked again at old prime number lists on the Encyclopedia of Integer Sequences.

Over eight year ago, I published a post on the 16,719^th century, the first to contain no prime number, and some other sequences with very few primes. Today, using the trusty Alpertron factoring site, I turned my attention to the 473,268^th and 473,269^th centuries — the first consecutive centuries void of prime numbers. The factors of numbers not divisible by 2, 3, or 5 in the 473,268^th and 473,269^th centuries are tabulated below:

• 47,326,703 = 43 × 73 × 15,077
• 47,326,709 = 23 × 2,057,683
• 47,326,711 = 331 × 142,981
• 47,326,717 = 2,939 × 16,103
• 47,326,721 = 13 × 53 × 149 × 461
• 47,326,723 = 67 × 706,369
• 47,326,727 = 7 × 233 × 29,017
• 47,326,729 = 89 × 643 × 827
• 47,326,733 = 2521 × 18,773
• 47,326,739 = 19 × 19 × 31 × 4,229
• 47,326,741 = 7 × 11 × 614,633
• 47,326,747 = 13 × 3,640,519
• 47,326,751 = 41 × 1,154,311
• 47,326,753 = 29 × 1,631,957
• 47,326,757 = 5,477 × 8,641
• 47,326,759 = 17 × 2,783,927
• 47,326,763 = 11 × 131 × 32,843
• 47,326,769 = 7 × 1,291 × 5,237
• 47,326,771 = 1,399 × 33,829
• 47,326,777 = 19 × 199 × 12,517
• 47,326,781 = 829 × 57,089
• 47,326,783 = 7 × 6,760,969
• 47,326,787 = 137 × 345,451
• 47,326,789 = 43 × 61 × 18,043
• 47,326,793 = 17 × 523 × 5,323
• 47,326,799 = 13 × 3,640,523
• 47,326,801 = 23 × 31 × 66,377
• 47,326,807 = 11 × 257 × 16,741
• 47,326,811 = 7 × 29 × 37 × 6,301
• 47,326,813 = 307 × 154,159
• 47,326,817 = 431 × 109,807
• 47,326,819 = 109 × 434,191
• 47,326,823 = 2,753 × 17,191
• 47,326,829 = 11 × 4,302,439
• 47,326,831 = 6,793 × 6,967
• 47,326,837 = 1,279 × 37,003
• 47,326,841 = 1,777 × 26,633
• 47,326,843 = 349 × 135,607
• 47,326,847 = 23 × 2,057,689
• 47,326,849 = 73 × 73 × 83 × 107
• 47,326,853 = 7 × 19 × 355,841
• 47,326,859 = 139 × 340,481
• 47,326,861 = 17 × 2,783,933
• 47,326,867 = 7 × 379 × 17,839
• 47,326,871 = 743 × 63,697
• 47,326,873 = 11 × 151 × 28,493
• 47,326,877 = 13 × 1,187 × 3,067
• 47,326,879 = 1,931 × 24,509
• 47,326,883 = 101 × 619 × 757
• 47,326,889 = 6,073 × 7,793
• 47,326,891 = 19 × 2,490,889
• 47,326,897 = 1,277 × 37,061

The square root of 47,326,900 is approximately 6,879.454920268029491. The largest prime smaller than the square root of 47,326,900 is 6,871, and 6,871 is the 872^nd prime number. If we tabulate all factors from the table above, we find that the following primes smaller than 6,879 divide numbers in the 473,268^th and 473,269^th centuries:

 

Prime	# of numbers divisible by
7	7
11	4
13	4
17	3
19	4
23	3
29	2
31	2
37	1
41	1
43	2
53	1
61	1
67	1
73	2
83	1
89	1
101	1
107	1
109	1
131	1
137	1
139	1
149	1
151	1
199	1
233	1
257	1
307	1
331	1
347	1
379	1
431	1
461	1
523	1
619	1
643	1
743	1
757	1
827	1
829	1
1,187	1
1,277	1
1,279	1
1,291	1
1,399	1
1,777	1
1,931	1
2,521	1
2,753	1
2,939	1
3,067	1
4,229	1
5,237	1
5,323	1
5,477	1
6,073	1
6,301	1
6,793	1

In other words, only fifty-nine primes out of 872 are needed to divide numbers in the first pair of consecutive primefree centuries (excluding number divisible by 2, 3, or 5 which may also have other factors smaller than 6,879). Put another way, just 6.53 percent of the primes that could potentially be the smallest factor of a number in the 473,268^th or 473,269^th centuries actually divide any number therewithin. With similar sequences in seven-digit centuries — just one order of magnitude smaller — the proportion of primes required is three times greater, which I find somewhat surprising. It obviously suggests that “pairing” of factors rapidly becomes more difficult as numbers become larger, so that it is more difficult for possible factors to occur in rapid succession as happens during the persistent record gap between 1,327 and 1,361.