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” 1,790,288th Century:
179028701 = 71 × 277 × 9103179028703 = 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.]
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 6,328th century [632713, 632717, 632743, 632747, 632773, 632777]
The 2,527,250th Century — 15 Primes, None Ending in 7:
252724901 = 11 × 13 × 17673071. 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.]
- six primes ending in 1
- five primes ending in 3
- no prime ending in 7
- 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 1,573rd 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