# examples of ranges of consecutive integers for Erdős-Woods numbers

The most famous example of 16 as an Erdős-Woods number (http://planetmath.org/ErdHosWoodsNumber) is the range of 16 consecutive integers starting with 2184.

Another $n$ for $k=16$ is 2044224, which we obtained by multiplying 2184 by 936. The factorization is $2044224={2}^{6}\times {3}^{3}\times 7\times {13}^{2}$, while $2044224+16=2044240={2}^{4}\times 5\times 11\times 23\times 101$. The table of factorizations

2044225 | ${5}^{2}\times 81769$ |
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2044226 | $2\times 1022113$ |

2044227 | $3\times 681409$ |

2044228 | ${2}^{2}\times 511057$ |

2044229 | $11\times 19\times 9781$ |

2044230 | $2\times 3\times 5\times 68141$ |

2044231 | ${7}^{2}\times 41719$ |

2044232 | ${2}^{3}\times 59\times 61\times 71$ |

2044233 | ${3}^{2}\times 17\times 31\times 431$ |

2044234 | $2\times 1009\times 1013$ |

2044235 | $5\times 107\times 3821$ |

2044236 | ${2}^{2}\times 3\times 170353$ |

2044237 | $13\times 67\times 2347$ |

2044238 | $2\times 7\times 151\times 967$ |

2044239 | $3\times 29\times 23497$ |

shows that each of the numbers in this range shares at least one factor with one if not both of the numbers capping the range.

Next we have a slightly longer example, this one for $k=34$. The smallest matching $n$ is 47563752566, a squarefree^{} number with a factorization of $2\times 11\times 17\times 23\times 41\times 157\times 859$. The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of ${2}^{3}\times {3}^{2}\times {5}^{2}\times 7\times 13\times 17\times 19\times 29\times 31$. While the size of these numbers forbids verification on your typical pocket calculator, these numbers are well within the reach of a Javascript implementation of trial division^{}. Here we could be tempted to omit the even numbers^{}, as they obviously share 2 as a prime factor^{} with the range start and the range end, as well as multiples^{} of 3 or 5 as they thus share factors with the range end. But, on the hope that it turns out to be at least a little bit instructive, the factorizations of all the numbers in our chosen range is given.

47563752567 | $3\times 3719\times 4263131$ |
---|---|

47563752568 | ${2}^{3}\times 71\times 199\times 420799$ |

47563752569 | $31\times 163\times 9412973$ |

47563752570 | $2\times 3\times 5\times 1585458419$ |

47563752571 | $29\times 12941\times 126739$ |

47563752572 | ${2}^{2}\times {7}^{2}\times 242672207$ |

47563752573 | ${3}^{2}\times 4657\times 1134821$ |

47563752574 | $2\times 13\times 823\times 991\times 2243$ |

47563752575 | ${5}^{2}\times 31769\times 59887$ |

47563752576 | ${2}^{7}\times 3\times 349\times 354911$ |

47563752577 | $11\times 397\times 593\times 18367$ |

47563752578 | $2\times 173\times 137467493$ |

47563752579 | $3\times 7\times 3257\times 695407$ |

47563752580 | ${2}^{2}\times 5\times 83\times 617\times 46439$ |

47563752581 | $19\times 2503355399$ |

47563752582 | $2\times {3}^{4}\times 53\times 59\times 93893$ |

47563752583 | $17\times 43\times 5171\times 12583$ |

47563752584 | ${2}^{3}\times 149\times 39902477$ |

47563752585 | $3\times 5\times 67\times 47327117$ |

47563752586 | $2\times 7\times 3397410899$ |

47563752587 | ${13}^{2}\times 281442323$ |

47563752588 | ${2}^{2}\times 3\times 11\times 4513\times 79843$ |

47563752589 | $23\times 61\times 151\times 224513$ |

47563752590 | $2\times 5\times 4756375259$ |

47563752591 | ${3}^{2}\times 5284861399$ |

47563752592 | ${2}^{4}\times 47\times 63249671$ |

47563752593 | $7\times 6794821799$ |

47563752594 | $2\times 3\times 7927292099$ |

47563752595 | $5\times 32503\times 292673$ |

47563752596 | ${2}^{2}\times 11890938149$ |

47563752597 | $3\times 15854584199$ |

47563752598 | $2\times 23781876299$ |

47563752599 | $11\times 37\times 127\times 373\times 2467$ |

Title | examples of ranges of consecutive integers for Erdős-Woods numbers |
---|---|

Canonical name | ExamplesOfRangesOfConsecutiveIntegersForErdHosWoodsNumbers |

Date of creation | 2013-03-22 17:38:16 |

Last modified on | 2013-03-22 17:38:16 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 5 |

Author | PrimeFan (13766) |

Entry type | Example |

Classification | msc 11A05 |