An Erdős-Woods number is an integer 
for which there is an integer 
such that every integer
in the interval ![[a,a+k]](/images/equations/Erdos-WoodsNumber/Inline3.svg)
has a common factor greater
than 1 with at least one endpoint. Equivalently,
 |
for 
,
1, ..., 
,
where 
denotes the greatest common divisor (Cégielski
et al. 2003, pp. 53-54). The number 
is the difference between the
endpoints, so the interval
contains 
integers.
The first Erdős-Woods numbers are 16, 22, 34, 36, 46, 56, 64, 66, 70, 76, ... (OEIS A059756). The smallest is 16, witnessed
by 
and 
.
Their prime factorizations are 
and 
. Each of the 17 integers
from 2184 through 2200 shares a prime factor with
one of these endpoints (Cégielski et al. 2003,
pp. 54, 58).
For every Erdős-Woods number 
, the integer 
has at least two distinct prime
factors (Cégielski et al. 2003, p. 55). There are infinitely
many Erdős-Woods numbers (Dowe 1989, p. 86).
