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.