Theorem: Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit

The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit:
:$872\,894, 872\,895, 872\,896, 872\,897, 872\,898$
{{OEIS|A192545}}

Proof:
Numbers ending in $0$, $2$, $4$, $6$ and $8$ are not prime because by Divisibility by 2 they are divisible by $2$.
Numbers ending in $0$ and $5$ are not prime because by Divisibility by 5 they are divisible by $5$.
Hence each of $872\,894$, $872\,895$, $872\,896$ and $872\,898$ remain composite when you change any of their digits except the last one.
So we inspect the prime factors of the following, only bothering to check the numbers ending in $1$, $3$, $7$ and $9$:
{{begin-eqn}}
{{eqn | l = 872\,891
      | r = 271 \times 3221
      | c = Not prime
}}
{{eqn | l = 872\,893
      | r = 7 \times 124699
      | c = Not prime
}}
{{eqn | l = 872\,897
      | r = 263 \times 3319
      | c = Not prime
}}
{{eqn | l = 872\,897
      | r = 17 \times 51347
      | c = Not prime
}}
{{end-eqn}}
All we need to do now is to inspect $872\,897$.
{{begin-eqn}}
{{eqn | l = 072\,897
      | r = 3 \times 11 \times 472
      | c = Not prime
}}
{{eqn | l = 172\,897
      | r = 41 \times 4217
      | c = Not prime
}}
{{eqn | l = 272\,897
      | r = 19 \times 53 \times 271
      | c = Not prime
}}
{{eqn | l = 372\,897
      | r = 3^3 \times 7 \times 1973
      | c = Not prime
}}
{{eqn | l = 472\,897
      | r = 37 \times 12781
      | c = Not prime
}}
{{eqn | l = 572\,897
      | r = 13 \times 127 \times 347
      | c = Not prime
}}
{{eqn | l = 672\,897
      | r = 3 \times 224299
      | c = Not prime
}}
{{eqn | l = 772\,897
      | r = 757 \times 1021
      | c = Not prime
}}
{{eqn | l = 972\,897
      | r = 3 \times 324299
      | c = Not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 802\,897
      | r = 53 \times 15149
      | c = Not prime
}}
{{eqn | l = 812\,897
      | r = 733 \times 1109
      | c = Not prime
}}
{{eqn | l = 822\,897
      | r = 3^2 \times 91433
      | c = Not prime
}}
{{eqn | l = 832\,897
      | r = 13 \times 79 \times 811
      | c = Not prime
}}
{{eqn | l = 842\,897
      | r = 11 \times 19 \times 37 \times 109
      | c = Not prime
}}
{{eqn | l = 852\,897
      | r = 3 \times 107 \times 2657
      | c = Not prime
}}
{{eqn | l = 862\,897
      | r = 7 \times 131 \times 941
      | c = Not prime
}}
{{eqn | l = 882\,897
      | r = 3 \times 151 \times 1949
      | c = Not prime
}}
{{eqn | l = 892\,897
      | r = 607 \times 1471
      | c = Not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 870\,897
      | r = 3 \times 61 \times 4759
      | c = Not prime
}}
{{eqn | l = 871\,897
      | r = 13 \times 47 \times 1427
      | c = Not prime
}}
{{eqn | l = 873\,897
      | r = 3 \times 291299
      | c = Not prime
}}
{{eqn | l = 874\,897
      | r = 23 \times 38039
      | c = Not prime
}}
{{eqn | l = 875\,897
      | r = 11 \times 79627
      | c = Not prime
}}
{{eqn | l = 876\,897
      | r = 3^2 \times 7 \times 31 \times 449
      | c = Not prime
}}
{{eqn | l = 877\,897
      | r = 17 \times 113 \times 457
      | c = Not prime
}}
{{eqn | l = 878\,897
      | r = 139 \times 6323
      | c = Not prime
}}
{{eqn | l = 879\,897
      | r = 3 \times 37 \times 7927
      | c = Not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 872\,097
      | r = 3 \times 149 \times 1951
      | c = Not prime
}}
{{eqn | l = 872\,197
      | r = 59 \times 14783
      | c = Not prime
}}
{{eqn | l = 872\,297
      | r = 191 \times 4567
      | c = Not prime
}}
{{eqn | l = 872\,397
      | r = 3^3 \times 79 \times 409
      | c = Not prime
}}
{{eqn | l = 872\,497
      | r = 37 \times 23581
      | c = Not prime
}}
{{eqn | l = 872\,597
      | r = 11 \times 23 \times 3449
      | c = Not prime
}}
{{eqn | l = 872\,697
      | r = 3 \times 7 \times 29 \times 1433
      | c = Not prime
}}
{{eqn | l = 872\,797
      | r = 17 \times 51341
      | c = Not prime
}}
{{eqn | l = 872\,997
      | r = 3 \times 290999
      | c = Not prime
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 872\,807
      | r = 13 \times 67139
      | c = Not prime
}}
{{eqn | l = 872\,817
      | r = 3 \times 11 \times 26449
      | c = Not prime
}}
{{eqn | l = 872\,827
      | r = 23 \times 137 \times 277
      | c = Not prime
}}
{{eqn | l = 872\,837
      | r = 7^2 \times 47 \times 379
      | c = Not prime
}}
{{eqn | l = 872\,847
      | r = 3^2 \times 293 \times 331
      | c = Not prime
}}
{{eqn | l = 872\,857
      | r = 43 \times 53 \times 383
      | c = Not prime
}}
{{eqn | l = 872\,867
      | r = 31 \times 37 \times 761
      | c = Not prime
}}
{{eqn | l = 872\,877
      | r = 3 \times 290959
      | c = Not prime
}}
{{eqn | l = 872\,887
      | r = 523 \times 1669
      | c = Not prime
}}
{{end-eqn}}
The result has been proven.
{{qed}}