“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.