Three strange fifteen-prime patterns

Looking through prime patterns today, I have come across two very interesting patterns for nine-digit fifteen-prime centuries, and one of six six-digit fifteen-prime centuries that has extraordinarily few possibilities to have so many:

“Full 9” 1,790,288^th Century:

179028701 = 71 × 277 × 9103

179028703 = 7 × 7 × 3653647

179028707 = 11 × 13 × 409 × 3061

1. 179028709 is prime [1.]

2. 179028713 is prime

3. 179028719 is prime [2.]

179028721 = 3373 × 53077

179028727 = 1699 × 105373

179028731 = 7 × 1163 × 21991

179028733 = 13 × 1451 × 9491

179028737 = 43 × 4163459

4. 179028739 is prime [3.]

5. 179028743 is prime

6. 179028749 is prime [4.]

179028751 = 11 × 17 × 31 × 89 × 347

179028757 = 23 × 67 × 116177

7. 179028761 is prime

8. 179028763 is prime

9. 179028767 is prime

10. 179028769 is prime [5.]

179028773 = 7 × 11 × 19 × 79 × 1549

11. 179028779 is prime [6.]

179028781 = 47 × 3809123

179028787 = 7 × 25575541 

12. 179028791 is prime

13. 179028793 is prime

14. 179028797 is prime

15. 179028799 is prime [7.]

Here we see a fifteen-prime century of the form 3k+1 with seven primes ending in 9 (a “full 9” century). The only smaller century with seven primes ending in 9 I know of is from 21,169,600 to 21,169,699, and it is not impossible that the 1,790,288^th century really is the second with seven primes ending in 9, though I have not asked about this yet.

What makes this pattern even stranger is that eight of the fifteen primes — six of the eight that do not end in 9 — are part of a pair of successive prime quadruples. The other two primes, both of which end in 3, also fit a remarkable pattern. In fact, except for 179,028,773, which has several small prime divisors and could be detected from the century’s modulus (231), there are only four decadal patterns for the whole century, each of which occurs multiple times. The only comparable pattern of which I am aware occurs in the fairly well-known 6,328^th century [632713, 632717, 632743, 632747, 632773, 632777]

The 2,527,250^th Century — 15 Primes, None Ending in 7:

252724901 = 11 × 13 × 1767307

1. 252724903 is prime [1.]

252724907 = 17 × 59 × 251969

252724909 = 19 × 19 × 179 × 3911

252724913 = 7 × 5197 × 6947

2. 252724919 is prime [1.]

3. 252724921 is prime [1.]

252724927 = 7 × 13 × 31 × 101 × 887

4. 252724931 is prime [2.]

5. 252724933 is prime [2.]

252724937 = 29 × 281 × 31013

6. 252724939 is prime [2.]

252724943 = 23 × 41 × 283 × 947

7. 252724949 is prime [3.]

8. 252724951 is prime [3.]

252724957 = 83 × 3044879

9. 252724961 is prime [4.]

10. 252724963 is prime [3.]

252724967 = 11 × 22974997

252724969 = 7 × 47 × 768161

11. 252724973 is prime [4.]

252724979 = 13 × 19440383

12. 252724981 is prime [5.]

252724987 = 1669 × 151423

13. 252724991 is prime [6.]

14. 252724993 is prime [5.]

252724997 = 7 × 7 × 71 × 72643

15. 252724999 is prime [4.]

What we see here is a fifteen-prime century with:

1. six primes ending in 1
2. five primes ending in 3
3. no prime ending in 7
4. four primes ending in 9

Although by deletion from several very small centuries with sixteen primes — the fourth and eleventh — one can easily draw up patterns with fourteen primes but no prime ending in one digit, this century with fifteen primes but not one ending in 7 surprised me so much I felt I had to tabulate it. What is strange here is that seven of thirteen composite numbers not divisible by 2, 3, or 5 end in 7.

The 1,573^rd Century — Almost All Potential Primes Prime

This century is much smaller than the previous two, and it is only today that I realised how improbable it was when one tests its numbers modulo not merely 3, but also 7, 11 and 13. Unlike the two larger centuries, I will tabulate all numbers not divisible by 2 or 5 to illustrate:

157201 = 11 × 31 × 461

157203 = 3 × 3 × 17467

1. 157207 is prime

157209 = 3 × 13 × 29 × 139

2. 157211 is prime

157213 = 7 × 37 × 607

3. 157217 is prime

4. 157219 is prime

157221 = 3 × 3 × 3 × 3 × 3 × 647

157223 = 11 × 14293

157227 = 3 × 7 × 7487

5. 157229 is prime

6. 157231 is prime

157233 = 3 × 17 × 3083

157237 = 97 × 1621

157239 = 3 × 3 × 17471

157241 = 7 × 7 × 3209

7. 157243 is prime

8. 157247 is prime

157249 = 67 × 2347

157251 = 3 × 23 × 43 × 53

9. 157253 is prime

157257 = 3 × 3 × 101 × 173

10. 157259 is prime

157261 = 13 × 12097

157263 = 3 × 19 × 31 × 89

157267 = 11 × 17 × 29 × 29

157269 = 3 × 7 × 7489

11. 157271 is prime

12. 157273 is prime

13. 157277 is prime

14. 157279 is prime

157281 = 3 × 103 × 509

157283 = 7 × 22469

157287 = 3 × 13 × 37 × 109

157289 = 11 × 79 × 181

15. 157291 is prime

157293 = 3 × 3 × 17477

157297 = 7 × 23 × 977

157299 = 3 × 52433

Here, we see that all but seventeen numbers, based on divisibility criteria, cannot be prime. Yet, of these seventeen, a remarkable fifteen actually are prime! Looking through tables modulo 21, 231 and finally 3003, I have never seen a case like this century. Despite the fact that nine numbers out of the 26 not divisible by 2, 3 or 5 can be ruled out modulo 7, 11 and/or 13, there are only two other composite numbers in the whole century. Although with centuries above the “eight-digit gap” I have seen this, in those cases all the composite numbers have highly complex factorisations, whereas 157,237 and 157,249 are semiprimes of a type familiar when learning early factorisations.