Consider a sequence of consecutive positive integers <math>[a, a+1, \dots a+k]</math>. The length k is an Erdős-Woods number if there exists such a sequence in which each of the elements has a common factor with one of the endpoints, i.e. if there exists a positive integer a such that for each integer i, <math>0 \le i \le k</math>, either <math>\gcd(a, a+i) > 1</math> or <math>\gcd(a+i, a+k) > 1</math>. The Erdős-Woods numbers are listed as (sequence A059756 in OEIS). The first few are given as
though arguably 0 and 1 could also be included as trivial entries. A059757 lists the starting point of the corresponding sequences. Investigation of such numbers stemmed from a prior conjecture by Paul Erdős:
- There exists a positive integer k such that every integer a is uniquely determined by the list of prime divisors of <math>a, a+1, \dots a+k</math>.
Alan R. Woods investigated this for his 1981 thesis, and conjectured that whenever k > 1, the interval <math>[a, a+k]</math> always included a number coprime to both endpoints. It was only later that he found the first counterexample, <math>[2184, 2185, \dots 2200]</math> with <math>k = 16</math>. David L. Dowe proved that there are infinitely many Erdős-Woods numbers, and Cégielski, Heroult and Richard showed that the set is recursive.
