Find a higher set of four prime numbers than 9431, 9433, 9437 and 9439 within ten consecutive numbers?

Question

At the most there can only be four prime numbers within ten consecutive numbers. Another example would be 101, 103, 107 and 109. I am looking for a larger set of numbers than shown in the question.

Answer 1

This is an easy question to answer using a computer program, so I wrote one.

Here is a list of all such sets of primes below 100,000:

2 3 5 7
3 5 7 11
5 7 11 13
11 13 17 19
101 103 107 109
191 193 197 199
821 823 827 829
1481 1483 1487 1489
1871 1873 1877 1879
2081 2083 2087 2089
3251 3253 3257 3259
3461 3463 3467 3469
5651 5653 5657 5659
9431 9433 9437 9439
13001 13003 13007 13009
15641 15643 15647 15649
15731 15733 15737 15739
16061 16063 16067 16069
18041 18043 18047 18049
18911 18913 18917 18919
19421 19423 19427 19429
21011 21013 21017 21019
22271 22273 22277 22279
25301 25303 25307 25309
31721 31723 31727 31729
34841 34843 34847 34849
43781 43783 43787 43789
51341 51343 51347 51349
55331 55333 55337 55339
62981 62983 62987 62989
67211 67213 67217 67219
69491 69493 69497 69499
72221 72223 72227 72229
77261 77263 77267 77269
79691 79693 79697 79699
81041 81043 81047 81049
82721 82723 82727 82729
88811 88813 88817 88819
97841 97843 97847 97849
99131 99133 99137 99139

Answer 2

Following are the higher set of prime no within 10 consecutive no:

Set-1

13001
13003
13007
13009

Set-2

16061
16063
16067
16069

set-3

15731
15733
15737
15739

set-4

18041
18043
18047
18049

set-5

19421
19423
19427
19429

Answer 3

as above