_{19}) to be prime.

Whilst I do not understand how Oscar Hoppe did it except by showing that R

_{19}could

*not*be expressed as the difference of nonconsecutive squares, I thought I should try and do something I had wanted to do many years ago. That is to see how many of the possible prime factors actually divide numbers in some major prime gaps or sequences with very few primes, in order to see what size prime gap would theoretically be possible if all “small” primes (defined as those under the square root) divided a number within a sequence of a numbers of an arbitrary size.

The sequences examined are:

- 31,398 to 31,468 – a highly persistent prime gap, the largest with five-digit numbers and the most persistent above 1,361
- 155,900 to 155,999 and 268,300 to 268,399 – the first two centuries with one prime only (the first with
*two*primes is actually from 302,000 to 302,099) - 370,262 to 370,372 and 370,900 to 371,020 – the first being the smallest prime gap of over 100 and the second the second century with only two primes.
- 1,357,202 to 1,357,332 – the first of two sequences of 131 composite numbers in the second million
- the 16,719
^{th}century – the first with no primes whatsoever – and surrounding prime-free numbers.

## 31398 to 31468

Number | “Small” Factors | “Large” Factors | ||
---|---|---|---|---|

31,399 | 17 | 1847 | ||

31,403 | 31 | 1013 | ||

31,409 | 7 | 7 | 641 | |

31,411 | 101 | 311 | ||

31,417 | 89 | 353 | ||

31,421 | 13 | 2417 | ||

31,423 | 7 | 67 | 67 | |

31,427 | 11 | 2857 | ||

31,429 | 53 | 593 | ||

31,433 | 17 | 43 | 43 | |

31,439 | 149 | 211 | ||

31,441 | 23 | 1367 | ||

31,447 | 13 | 41 | 59 | |

31,451 | 7 | 4493 | ||

31,453 | 71 | 443 | ||

31,457 | 83 | 379 | ||

31,459 | 163 | 193 | ||

31,463 | 73 | 431 |

Total number of prime factors under square root required: 18 of 38 (47.36 percent).

## 58789 to 58888

Number | “Small” Factors | “Large” Factors | ||
---|---|---|---|---|

58,793 | 7 | 37 | 227 | |

58,799 | 13 | 4523 | ||

58,801 | 127 | 463 | ||

58,807 | 7 | 31 | 271 | |

58,811 | 23 | 2557 | ||

58,813 | 103 | 571 | ||

58,817 | 11 | 5347 | ||

58,819 | 131 | 449 | ||

58,823 | 59 | 997 | ||

58,829 | 89 | 661 | ||

58,831 |
is a prime
| |||

58,837 | 17 | 3461 | ||

58,841 | 29 | 2029 | ||

58,843 | 19 | 19 | 163 | |

58,847 | 83 | 709 | ||

58,849 | 7 | 7 | 1201 | |

58,853 | 229 | 257 | ||

58,859 | 71 | 829 | ||

58,861 | 11 | 5351 | ||

58,867 | 37 | 1591 | ||

58,871 | 17 | 3463 | ||

58,873 | 113 | 521 | ||

58,877 | 7 | 13 | 647 | |

58,879 | 97 | 607 | ||

58,883 | 11 | 53 | 101 |

Total number of prime factors under square root required: 23 of 46 (50.00 percent).

## 155,894 to 156,001

Number | “Small” Factors | “Large” Factors | ||
---|---|---|---|---|

155,897 | 7 | 22271 | ||

155,899 | 31 | 47 | 107 | |

155,903 | 11 | 14173 | ||

155,909 | 13 | 67 | 179 | |

155,911 | 7 | 22273 | ||

155,917 | 23 | 6779 | ||

155,921 |
is a prime
| |||

155,923 | 41 | 3803 | ||

155,927 | 241 | 647 | ||

155,929 | 211 | 739 | ||

155,933 | 19 | 29 | 283 | |

155,939 | 7 | 22277 | ||

155,941 | 17 | 9173 | ||

155,947 | 11 | 14177 | ||

155,951 | 277 | 563 | ||

155,953 | 7 | 22279 | ||

155,957 | 83 | 1879 | ||

155,959 | 263 | 593 | ||

155,963 | 23 | 6781 | ||

155,969 | 11 | 11 | 1289 | |

155,971 | 19 | 8209 | ||

155,977 | 61 | 2557 | ||

155,981 | 7 | 22283 | ||

155,983 | 151 | 1033 | ||

155,987 | 13 | 13 | 923 | |

155,989 | 389 | 401 | ||

155,993 | 47 | 3319 | ||

155,999 | 257 | 607 | ||

156,001 | 73 | 2137 |

Total number of prime factors under square root required: 23 of 74 (31.08 percent).

##
2684^{th} century

Number | “Small” Factors | “Large” Factors | ||
---|---|---|---|---|

268,301 | 11 | 24391 | ||

268,303 | 7 | 38329 | ||

268,307 | 13 | 20639 | ||

268,309 | 71 | 3779 | ||

268,313 | 157 | 1709 | ||

268,319 | 251 | 1069 | ||

268,321 | 179 | 1499 | ||

268,327 | 151 | 1777 | ||

268,331 | 7 | 38333 | ||

268,333 | 13 | 20641 | ||

268,337 | 19 | 29 | 487 | |

268,339 | 53 | 61 | 83 | |

268,343 |
is a prime
| |||

268,349 | 149 | 1801 | ||

268,351 | 127 | 2113 | ||

268,357 | 101 | 2657 | ||

268,361 | 37 | 7253 | ||

268,363 | 43 | 79 | 79 | |

268,367 | 11 | 31 | 787 | |

268,369 | 167 | 1607 | ||

268,373 | 7 | 7 | 5477 | |

268,379 | 17 | 15787 | ||

268,381 | 349 | 769 | ||

268,387 | 7 | 23 | 1667 | |

268,391 | 59 | 4549 | ||

268,393 | 311 | 863 | ||

268,397 | 239 | 1123 | ||

268,399 | 97 | 2767 |

Total number of prime factors under square root required: 29 of 95 (30.52 percent).

## First century gap

Number | “Small” Factors | “Large” Factors | |||
---|---|---|---|---|---|

370,267 | 479 | 773 | |||

370,271 | 11 | 41 | 821 | ||

370,273 | 43 | 79 | 109 | ||

370,277 | 17 | 23 | 947 | ||

370,279 | 7 | 13 | 13 | 313 | |

370,283 | 379 | 977 | |||

370,289 | 349 | 1061 | |||

370,291 | 19 | 19489 | |||

370,297 | 353 | 1049 | |||

370,301 | 29 | 113 | 113 | ||

370,303 | 367 | 1009 | |||

370,307 | 7 | 52901 | |||

370,309 | 67 | 5527 | |||

370,313 | 47 | 7879 | |||

370,319 | 547 | 677 | |||

370,321 | 7 | 52903 | |||

370,327 | 107 | 3461 | |||

370,331 | 13 | 61 | 467 | ||

370,333 | 37 | 10009 | |||

370,337 | 11 | 131 | 257 | ||

370,339 | 199 | 1861 | |||

370,343 | 59 | 6277 | |||

370,349 | 7 | 191 | 277 | ||

370,351 | 179 | 2069 | |||

370,357 | 13 | 31 | 919 | ||

370,361 | 383 | 967 | |||

370,363 | 7 | 157 | 337 | ||

370,367 | 19 | 101 | 193 | ||

370,369 | 23 | 16103 |

Total number of prime factors under square root required: 38 of 108 (35.18 percent).

##
3,710^{th} and early 3,711^{st} centuries

Number | “Small” Factors | “Large” Factors | |||
---|---|---|---|---|---|

370,901 | 421 | 881 | |||

370,903 | 13 | 103 | 277 | ||

370,907 | 167 | 2221 | |||

370,909 | 7 | 11 | 4817 | ||

370,913 | 73 | 5081 | |||

370,919 |
is a prime
| ||||

370,921 | 23 | 16127 | |||

370,927 | 41 | 9047 | |||

370,931 | 11 | 33721 | |||

370,933 | 59 | 6287 | |||

370,937 | 7 | 19 | 2789 | ||

370,939 | 29 | 12791 | |||

370,943 | 347 | 1069 | |||

370,949 |
is a prime
| ||||

370,951 | 7 | 197 | 269 | ||

370,957 | 17 | 21821 | |||

370,961 | 43 | 8627 | |||

370,963 | 409 | 907 | |||

370,967 | 23 | 127 | 127 | ||

370,969 | 107 | 3467 | |||

370,973 | 101 | 3673 | |||

370,979 | 7 | 7 | 67 | 113 | |

370,981 | 13 | 28537 | |||

370,987 | 349 | 1063 | |||

370,991 | 17 | 139 | 157 | ||

370,993 | 7 | 52999 | |||

370,997 | 11 | 29 | 1163 | ||

370,999 | 37 | 37 | 271 | ||

371,003 | 353 | 1051 | |||

371,009 | 41 | 9049 | |||

371,011 | 577 | 643 | |||

371,017 | 563 | 659 | |||

371,021 | 7 | 53003 | |||

371,023 | 311 | 1193 |

Total number of prime factors under square root required: 33 of 108 (30.56 percent).

##
13,573^{rd} century

Number | “Small” Factors | “Large” Factors | ||
---|---|---|---|---|

1,357,207 | 23 | 59009 | ||

1,357,211 | 31 | 43781 | ||

1,357,213 | 11 | 13 | 9491 | |

1,357,217 | 257 | 5281 | ||

1,357,219 | 47 | 67 | 431 | |

1,357,223 | 7 | 41 | 4729 | |

1,357,229 | 17 | 29 | 2753 | |

1,357,231 | 103 | 13177 | ||

1,357,237 | 7 | 193891 | ||

1,357,241 | 149 | 9109 | ||

1,357,243 | 113 | 12011 | ||

1,357,247 | 307 | 4421 | ||

1,357,249 | 127 | 10687 | ||

1,357,253 | 23 | 59011 | ||

1,357,259 | 137 | 9907 | ||

1,357,261 | 349 | 3889 | ||

1,357,267 | 571 | 2377 | ||

1,357,271 | 37 | 36683 | ||

1,357,273 | 31 | 43783 | ||

1,357,277 | 53 | 25609 | ||

1,357,279 | 7 | 11 | 17627 | |

1,357,283 | 229 | 5927 | ||

1,357,289 | 73 | 18593 | ||

1,357,291 | 13 | 131 | 797 | |

1,357,297 | 17 | 79841 | ||

1,357,301 | 11 | 163 | 757 | |

1,357,303 | 19 | 71437 | ||

1,357,307 | 7 | 71 | 2731 | |

1,357,309 | 727 | 1867 | ||

1,357,313 | 47 | 28879 | ||

1,357,319 | 181 | 7499 | ||

1,357,321 | 7 | 97 | 1999 | |

1,357,327 | 449 | 3023 | ||

1,357,331 | 17 | 79843 |

Total number of prime factors under square root required: 33 of 189 (17.46 percent).

##
16,719^{th} century and surrounds

Number | “Small” Factors | “Large” Factors | |||
---|---|---|---|---|---|

1,671,787 | 13 | 128599 | |||

1,671,791 | 11 | 19 | 19 | 421 | |

1,671,793 | 103 | 16231 | |||

1,671,797 | 17 | 43 | 2287 | ||

1,671,799 | 31 | 199 | 271 | ||

1,671,803 | 7 | 238829 | |||

1,671,809 | 599 | 2791 | |||

1,671,811 | 137 | 12203 | |||

1,671,817 | 7 | 241 | 991 | ||

1,671,821 | 29 | 57649 | |||

1,671,823 | 191 | 8753 | |||

1,671,827 | 61 | 27407 | |||

1,671,829 | 19 | 87991 | |||

1,671,833 | 1289 | 1297 | |||

1,671,839 | 13 | 128603 | |||

1,671,841 | 1223 | 1367 | |||

1,671,847 | 23 | 72689 | |||

1,671,851 | 67 | 24953 | |||

1,671,853 | 101 | 16553 | |||

1,671,857 | 11 | 11 | 41 | 337 | |

1,671,859 | 7 | 238837 | |||

1,671,863 | 359 | 4657 | |||

1,671,869 | 83 | 20143 | |||

1,671,871 | 487 | 3433 | |||

1,671,877 | 79 | 21163 | |||

1,671,881 | 331 | 5051 | |||

1,671,883 | 43 | 59 | 659 | ||

1,671,887 | 7 | 238841 | |||

1,671,889 | 521 | 3209 | |||

1,671,893 | 23 | 157 | 463 | ||

1,671,899 | 17 | 98347 | |||

1,671,901 | 7 | 11 | 21713 |

Total number of prime factors under square root required: 36 of 207 (17.39 percent).

It is clearly noticeable how, as our numbers become bigger and bigger, the proportion of primes under the square root required to factor long sequences of composite numbers becomes smaller and smaller.

In the sequence from 1,328 to 1,360,

*every prime less than the square root*is used to factor a number which cannot be proven composite by elementary divisibility tests for 2, 3 and 5. In the 16,719

^{th}century, the smallest devoid of primes, only 33 primes of 207 are needed to do factor 24 numbers. This result from the 16,719

^{th}century and similar cases with smaller numbers, suggests that if every possible factor had a multiple of a prime larger than the square root, then gaps of over one hundred could occur easily with

*five*-digit numbers.

The “accuracy” required to match small and large prime factors for the largest possible sequence of composite numbers – as happens in the fourteenth century – however, must be quite impossible to replicate even with six-digit numbers, owing to the fact that necessary factors are often much larger than those used from 1,328 to 1,360. Hence, perhaps the observed gaps are more impressive than the actual use of small prime factors would suggest.

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