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 for is 2044224, which we obtained by multiplying 2184 by 936. The factorization is , while . The table of factorizations
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 . The smallest matching is 47563752566, a squarefree number with a factorization of . The number capping the end of the range is the decidedly non-squarefree 47563752600, with a factorization of . 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.