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.

Comparing the prime-rich centuries: Part II

In my previous post, I compared the first few seventeen- and eighteen-prime centuries to see what their patterns were like.

Most of the smallest seventeen- and eighteen-prime centuries, as noted in that post, are of the form 100k to 100k+99 where k is a number of the form 3n+1. This is not unexpected. A century from 300n+100 to 300n+199 contains 28 numbers not divisible by 2, 3, or 5, whereas a century from 300n to 300n+99 or from 300n+200 to 300n+299 contains only 26 numbers not divisible by 2, 3, or 5. (Divisibility by 7, however, means no century can contain so many as 26 prime numbers: indeed, the maximum number of primes any century after the first can theoretically contain is 23, and no century larger than the second is known to contain more than 20 primes).

As noted in the previous post about prime-rich centuries, the second seventeen- and the second eighteen-prime centuries have k of the form 3n+2. The next seventeen-prime century with k of the form 3n+2 or 3n is the sixteenth overall and the first seventeen-prime century after than the smallest eighteen-prime one. This century — from 190,818,931,155,800 to 190,818,931,155,899 — is also the first century after 1,400 to 1,499 with a single-digit count of composite numbers not divisible by 2, 3, or 5.

Below the second eighteen-prime century from 2,335,286,971,401,800 to 2,335,286,971,401,899 (actually alongside the prime number 2,335,286,971,401,799 there are nineteen primes in 101 numbers) there are:

• five centuries with seventeen primes and k of the form 3n+2
• six centuries with seventeen primes and k of the form 3n

I will tabulate these twelve centuries as two tables of six centuries each, and compare their prime patterns as I did for the centuries with k of the form 3n+1.

First Five Seventeen-Prime Centuries and First Eighteen-Prime Century with k of form 3n+2

14	1908189311558	6157376214122	18883149995801	22930638581651	23352869714018
	190,818,931,155,803	615,737,621,412,203	1,888,314,999,580,103	2,293,063,858,165,103	2,335,286,971,401,803
1,409		615,737,621,412,209		2,293,063,858,165,109	2,335,286,971,401,809
		615,737,621,412,211		2,293,063,858,165,111
	190,818,931,155,817	615,737,621,412,217		2,293,063,858,165,117
	190,818,931,155,821			2,293,063,858,165,121	2,335,286,971,401,821
1,423	190,818,931,155,823	615,737,621,412,223	1,888,314,999,580,123	2,293,063,858,165,123	2,335,286,971,401,823
1,427		615,737,621,412,227	1,888,314,999,580,127		2,335,286,971,401,827
1,429		615,737,621,412,229	1,888,314,999,580,129		2,335,286,971,401,829
1,433	190,818,931,155,833	615,737,621,412,233	1,888,314,999,580,133	2,293,063,858,165,133
1,439		615,737,621,412,239		2,293,063,858,165,139
	190,818,931,155,841			2,293,063,858,165,141	2,335,286,971,401,841
1,447		615,737,621,412,247	1,888,314,999,580,147	2,293,063,858,165,147	2,335,286,971,401,847
1,451	190,818,931,155,851		1,888,314,999,580,151		2,335,286,971,401,851
1,453		615,737,621,412,253	1,888,314,999,580,153	2,293,063,858,165,153
	190,818,931,155,857
1,459	190,818,931,155,859		1,888,314,999,580,159	2,293,063,858,165,159	2,335,286,971,401,859
	190,818,931,155,863				2,335,286,971,401,863
	190,818,931,155,869	615,737,621,412,269	1,888,314,999,580,169	2,293,063,858,165,169	2,335,286,971,401,869
1,471	190,818,931,155,871	615,737,621,412,271	1,888,314,999,580,171		2,335,286,971,401,871
				2,293,063,858,165,177	2,335,286,971,401,877
1,481		615,737,621,412,281	1,888,314,999,580,181	2,293,063,858,165,181
1,483	190,818,931,155,883	615,737,621,412,283	1,888,314,999,580,183		2,335,286,971,401,883
1,487	190,818,931,155,887		1,888,314,999,580,187	2,293,063,858,165,187	2,335,286,971,401,887
1,489	190,818,931,155,889	615,737,621,412,289	1,888,314,999,580,189	2,293,063,858,165,189	2,335,286,971,401,889
1,493	190,818,931,155,893		1,888,314,999,580,193
1,499	190,818,931,155,899	615,737,621,412,299	1,888,314,999,580,199		2,335,286,971,401,899

Here, there seems more variation in the patterns than for the more numerous seventeen-prime centuries of the form 3n+1. Nevertheless, there is the remarkable century from 1,888,314,999,580,100 to 1,888,314,999,580,199 that has fifteen primes with identical last two digits to primes between 1,400 and 1,499! Given that there are 2,829,786 possible prime patterns for a century with seventeen primes, and that no repeating patterns with 17 primes are yet known, this is extraordinary since the larger century is merely the fifty-fifth containing seventeen prime numbers. At least one repeating century with fifteen primes is known, although I remain unaware of its identity, and it certainly does not occur as early as the 55^th century with 15 primes [856,019,600 to 856,019,699].

First Six Seventeen-Prime Centuries with k of form 3n

7658205745776	8078877131667	10137710652198	13862924841999	17176990713081	19441702516473
765,820,574,577,601	807,887,713,166,701		1,386,292,484,199,901
765,820,574,577,607	807,887,713,166,707	1,013,771,065,219,807	1,386,292,484,199,907		1,944,170,251,647,307
	807,887,713,166,711	1,013,771,065,219,811	1,386,292,484,199,911	1,717,699,071,308,111	1,944,170,251,647,311
765,820,574,577,613	807,887,713,166,713	1,013,771,065,219,813	1,386,292,484,199,913	1,717,699,071,308,113	1,944,170,251,647,313
		1,013,771,065,219,817			1,944,170,251,647,317
	807,887,713,166,719	1,013,771,065,219,819	1,386,292,484,199,919	1,717,699,071,308,119
765,820,574,577,623		1,013,771,065,219,823	1,386,292,484,199,923	1,717,699,071,308,123
765,820,574,577,629			1,386,292,484,199,929		1,944,170,251,647,329
	807,887,713,166,731	1,013,771,065,219,831		1,717,699,071,308,131
765,820,574,577,637		1,013,771,065,219,837	1,386,292,484,199,937	1,717,699,071,308,137
	807,887,713,166,741	1,013,771,065,219,841	1,386,292,484,199,941	1,717,699,071,308,141	1,944,170,251,647,341
			1,386,292,484,199,943		1,944,170,251,647,343
765,820,574,577,647				1,717,699,071,308,147	1,944,170,251,647,347
765,820,574,577,649	807,887,713,166,749	1,013,771,065,219,849		1,717,699,071,308,149
	807,887,713,166,753	1,013,771,065,219,853		1,717,699,071,308,153	1,944,170,251,647,353
765,820,574,577,659	807,887,713,166,759	1,013,771,065,219,859			1,944,170,251,647,359
765,820,574,577,661				1,717,699,071,308,161	1,944,170,251,647,361
	807,887,713,166,767		1,386,292,484,199,967	1,717,699,071,308,167	1,944,170,251,647,367
765,820,574,577,671	807,887,713,166,771		1,386,292,484,199,971
765,820,574,577,673		1,013,771,065,219,873		1,717,699,071,308,173	1,944,170,251,647,373
765,820,574,577,677	807,887,713,166,777	1,013,771,065,219,877	1,386,292,484,199,977	1,717,699,071,308,177
765,820,574,577,679		1,013,771,065,219,879	1,386,292,484,199,979	1,717,699,071,308,179
765,820,574,577,683	807,887,713,166,783		1,386,292,484,199,983	1,717,699,071,308,183	1,944,170,251,647,383
	807,887,713,166,789	1,013,771,065,219,889	1,386,292,484,199,989		1,944,170,251,647,389
765,820,574,577,691	807,887,713,166,791		1,386,292,484,199,991	1,717,699,071,308,191	1,944,170,251,647,391
765,820,574,577,697	807,887,713,166,797	1,013,771,065,219,897			1,944,170,251,647,397

This list seems, on the whole, more random than either list previously considered. No pair of these six centuries has more than thirteen common primes (807,887,713,166,700 and 1,013,771,065,219,800), which does not seem surprising. There is:

• one pair of last two digits (k13) that is prime for all six
• the same as for the first ten centuries with eighteen primes and ten composites not divisible by 2, 3, or 5
• one pair of last two digits (k43) with only two primes
• the minimum in the centuries mentioned in the preceding point is three