TITLE: Is there at least one prime between $n \times 100$ and $(n \times 100) + 100$ where $n \in \mathbb{N}$
QUESTION [5 upvotes]: Is there at least one prime between the number $n \times 100$ and $(n \times 100) + 100$ for any $n \in \mathbb{N}$ that can be $0$ ?
Question originally formulated by one of my friends.

REPLY [2 votes]: Earliest failure:
 1671800 = 2^3 * 5^2 * 13 * 643
 1671801 = 3 * 23 * 24229
 1671802 = 2 * 11 * 75991
 1671803 = 7 * 238829
 1671804 = 2^2 * 3^2 * 46439
 1671805 = 5 * 239 * 1399
 1671806 = 2 * 769 * 1087
 1671807 = 3 * 557269
 1671808 = 2^7 * 37 * 353
 1671809 = 599 * 2791
 1671810 = 2 * 3 * 5 * 7 * 19 * 419
 1671811 = 137 * 12203
 1671812 = 2^2 * 417953
 1671813 = 3^3 * 11 * 13 * 433
 1671814 = 2 * 17 * 49171
 1671815 = 5 * 334363
 1671816 = 2^3 * 3 * 41 * 1699
 1671817 = 7 * 241 * 991
 1671818 = 2 * 835909
 1671819 = 3 * 557273
 1671820 = 2^2 * 5 * 83591
 1671821 = 29 * 57649
 1671822 = 2 * 3^2 * 131 * 709
 1671823 = 191 * 8753
 1671824 = 2^4 * 7 * 11 * 23 * 59
 1671825 = 3 * 5^2 * 22291
 1671826 = 2 * 13 * 64301
 1671827 = 61 * 27407
 1671828 = 2^2 * 3 * 127 * 1097
 1671829 = 19 * 87991
 1671830 = 2 * 5 * 31 * 5393
 1671831 = 3^2 * 7^2 * 17 * 223
 1671832 = 2^3 * 53 * 3943
 1671833 = 1289 * 1297
 1671834 = 2 * 3 * 278639
 1671835 = 5 * 11 * 113 * 269
 1671836 = 2^2 * 417959
 1671837 = 3 * 47 * 71 * 167
 1671838 = 2 * 7 * 119417
 1671839 = 13 * 128603
 1671840 = 2^5 * 3^5 * 5 * 43
 1671841 = 1223 * 1367
 1671842 = 2 * 109 * 7669
 1671843 = 3 * 557281
 1671844 = 2^2 * 417961
 1671845 = 5 * 7 * 37 * 1291
 1671846 = 2 * 3 * 11 * 73 * 347
 1671847 = 23 * 72689
 1671848 = 2^3 * 17 * 19 * 647
 1671849 = 3^2 * 431^2
 1671850 = 2 * 5^2 * 29 * 1153
 1671851 = 67 * 24953
 1671852 = 2^2 * 3 * 7 * 13 * 1531
 1671853 = 101 * 16553
 1671854 = 2 * 835927
 1671855 = 3 * 5 * 227 * 491
 1671856 = 2^4 * 104491
 1671857 = 11^2 * 41 * 337
 1671858 = 2 * 3^2 * 293 * 317
 1671859 = 7 * 238837
 1671860 = 2^2 * 5 * 179 * 467
 1671861 = 3 * 31 * 17977
 1671862 = 2 * 835931
 1671863 = 359 * 4657
 1671864 = 2^3 * 3 * 69661
 1671865 = 5 * 13 * 17^2 * 89
 1671866 = 2 * 7 * 119419
 1671867 = 3^3 * 19 * 3259
 1671868 = 2^2 * 11 * 37997
 1671869 = 83 * 20143
 1671870 = 2 * 3 * 5 * 23 * 2423
 1671871 = 487 * 3433
 1671872 = 2^6 * 151 * 173
 1671873 = 3 * 7 * 79613
 1671874 = 2 * 835937
 1671875 = 5^6 * 107
 1671876 = 2^2 * 3^2 * 46441
 1671877 = 79 * 21163
 1671878 = 2 * 13 * 64303
 1671879 = 3 * 11 * 29 * 1747
 1671880 = 2^3 * 5 * 7^2 * 853
 1671881 = 331 * 5051
 1671882 = 2 * 3 * 17 * 37 * 443
 1671883 = 43 * 59 * 659
 1671884 = 2^2 * 47 * 8893
 1671885 = 3^2 * 5 * 53 * 701
 1671886 = 2 * 19 * 43997
 1671887 = 7 * 238841
 1671888 = 2^4 * 3 * 61 * 571
 1671889 = 521 * 3209
 1671890 = 2 * 5 * 11 * 15199
 1671891 = 3 * 13 * 163 * 263
 1671892 = 2^2 * 31 * 97 * 139
 1671893 = 23 * 157 * 463
 1671894 = 2 * 3^3 * 7 * 4423
 1671895 = 5 * 334379
 1671896 = 2^3 * 103 * 2029
 1671897 = 3 * 181 * 3079
 1671898 = 2 * 41 * 20389
 1671899 = 17 * 98347
 1671900 = 2^2 * 3 * 5^2 * 5573