Prime numbers

   

Within each interval [10n; 10n+9], there can be up to 4 prime numbers. They are known as prime quadruplet.
It is not known if there are infinitely many prime quadruplets.

What if you now consider intervals of the form [100n; 100n+99]?

Between 0 and 99, there are 25 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Between 100 and 199, there are 21 prime numbers: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197 and 199.

And then?

   

Within each interval [100n; 100n+99] with n > 1, there are 17 prime numbers or less…

…until you reach the interval [122,853,771,370,900; 122,853,771,370,999] (~1.2x1014) which contains 18 prime numbers: 122,853,771,370,900 + x, with x inside {1; 3; 7; 19; 21; 27; 31; 33; 37; 49; 51; 61; 69; 73; 87; 91; 97; 99}.

Quite an unexpected result, isn’t it?

   

The next interval with the same property is [2,335,286,971,401,800; 2,335,286,971,401,899] (~2.3x1015) which contains the following prime numbers: 2,335,286,971,401,800 + x with x inside {3; 9; 21; 23; 27; 29; 41; 47; 51; 59; 63; 69; 71; 77; 83; 87; 89; 99}.

You may note that the numbers within this interval are almost 20 times bigger than the ones within the previous interval (2.3x1015 vs. 1.2x1014).

The third one is [2,870,323,747,426,600; 2,870,323,747,426,699] (~2.9x1015): 1, 7, 9, 13, 21, 27, 33, 43, 51, 57, 61, 69, 73, 79, 87, 91, 93 and 99.

Then we have:

[14,478,586,548,170,200; 14,478,586,548,170,299] (~1.4x1016): 3, 9, 19, 21, 31, 37, 43, 49, 57, 63, 67, 73, 79, 81, 87, 91, 93 and 99.

[16,139,492,396,644,900; 16,139,492,396,644,999] (~1.6x1016): 1, 7, 9, 13, 19, 21, 31, 33, 37, 43, 51, 61, 63, 73, 79, 93, 97 and 99.

[16,897,570,820,963,800; 16,897,570,820,963,899] (~1.7x1016): 1, 3, 9, 21, 27, 39, 43, 49, 57, 63, 67, 73, 79, 81, 87, 91, 93 and 97.

[17,474,880,906,689,800; 17,474,880,906,689,899] (~1.7x1016): 3, 9, 19, 21, 27, 31, 37, 39, 49, 51, 61, 67, 73, 79, 87, 91, 93 and 97.

[20,755,224,123,135,700; 20,755,224,123,135,799] (~2.1x1016): 3, 7, 9, 13, 21, 27, 37, 39, 51, 63, 67, 69, 73, 79, 81, 87, 91 and 97.

[27,821,517,920,553,100; 27,821,517,920,553,199] (~2.8x1016): 1, 3, 7, 9, 21, 27, 31, 43, 51, 57, 63, 67, 69, 73, 87, 91, 93 and 97.

[31,230,332,890,972,000; 31,230,332,890,972,099] (~3.1x1016): 1, 3, 9, 13, 21, 27, 31, 43, 51, 57, 63, 69, 73, 79, 87, 91, 97 and 99.

[59,224,898,214,387,700 - 59,224,898,214,387,799] (~5.9x1016): 1, 7, 9, 13, 21, 39, 43, 49, 57, 61, 67, 69, 79, 81, 91, 93, 97 and 99.

Is this list finite or infinite?
Is it the last interval?
Will you find the next one?