An expected confirmation

In my previous post, I noted that although repeating centuries with fifteen primes are known (although I cannot discover their identity at present) I was sure that they would not occur as early as the 55^th century with fifteen primes. I discussed this in the context of a repeat of fifteen of seventeen primes from the century from 1,400 to 1,499 in the 55^th century with seventeen primes (from 1,888,314,999,580,100 to 1,888,314,999,580,199). For both these centuries primes form by adding all numbers within the set {23, 27, 29, 33, 47, 51, 53, 59, 71, 81, 83, 87, 89, 93, 99}.

Tonight I have actually computed the prime patterns for all centuries with fifteen primes less than one billion. There are 58 centuries smaller than one billion with fifteen primes, and fifteen is the largest number of primes in any century within the “eight-digit gap”, which extends from slightly below three million to 839,296,299. (This “eight-digit gap” is partly predictable from heuristics. Around this digit count the rate at which the probability of a century containing more the fifteen primes decreases equals the inverse of the rate at which the number of centuries increases, although the lowest theoretical probabilities of the existence of any century with sixteen or seventeen primes actually occur for seven digits).

As I strongly expected, none of the first fifty-eight centuries with fifteen primes have the same pattern. In fact, of the 1,653 possible pairs from these first fifty-eight centuries with fifteen primes, no pair forms common primes by adding a set larger than twelve numbers, and only four have this number of common primes.

It is too cumbersome and impractical to list the patterns of all 58 centuries; however, I will list the number of common primes for each pair:

# same primes	# centuries to 1 billion
0	0
1	15
2	66
3	147
4	282
5	260
6	190
7	196
8	205
9	170
10	96
11	22
12	4
13	0
14	0
15	0

The four pairs of centuries with twelve common primes are, in order of smallest larger century:

1. 6,300 to 6,399 and 2,967,300 to 2,967,399 with common primes formed by adding {17, 23, 29, 37, 43, 53, 59, 61, 73, 79, 89, 97}
2. 46,497,700 to 46,497,799 and 593,131,900 to 593,131,999 with common primes formed by adding {7, 9, 19, 27, 33, 39, 51, 61, 67, 91, 93, 97}
3. 516,257,800 to 516,257,899 and 738,740,200 to 738,740,299 with common primes formed by adding {1, 3, 9, 13, 21, 51, 57, 69, 79, 81, 91, 93}
4. 2,600 to 2,699 and 925,594,400 to 925,594,499 with common primes formed by adding {9, 33, 47, 57, 59, 63, 71, 77, 87, 89, 93, 99}
• both these last two are part of extremely prime-rich sequences that overlap two centuries:
• the century from 2,650 to 2,750 contains eighteen primes
• the century between 925,594,420 and 925,594,520 contains seventeen primes
• in fact all these seventeen primes are between 925,594,429 and 925,594,513 (85 numbers)
• yet, there is no case of so many as seventeen primes between consecutive multiples of 100 amongst nine- or even ten-digit numbers

Overall, there is nothing amongst the fifteen-prime centuries less than one billion to compare with the similarity of the prime patterns of the centuries beginning with 1,400 and 1,888,314,999,580,100. This is what I expected. Even taking into account that there are more than four times as many possible prime patterns for a fifteen-prime as for a seventeen-prime century, I had no expectations of something closer to a repeating century than the four cases mentioned above.