In number theory, a positive integer k is said to be an Erdős–Woods number if it has the following property: there exists a positive integer a such that in the sequence (a, a + 1, …, a + k) of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, k is an Erdős–Woods number if there exists a positive integer a such that for each integer i between 0 and k, at least one of the greatest common divisors gcd(a, a + i) and gcd(a + i, a + k) is greater than 1.
Contents
Examples
The first few Erdős–Woods numbers are
16, 22, 34, 36, 46, 56, 64, 66, 70 … (sequence A059756 in the OEIS).(Arguably 0 and 1 could also be included as trivial entries.)
History
Investigation of such numbers stemmed from the following 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 a, a + 1, …, a + k.Alan R. Woods investigated this question for his 1981 thesis. Woods conjectured that whenever k > 1, the interval [a, a + k] always includes a number coprime to both endpoints. It was only later that he found the first counterexample, [2184, 2185, …, 2200], with k = 16. The existence of this counterexample shows that 16 is an Erdős–Woods number.
Dowe (1989) proved that there are infinitely many Erdős–Woods numbers, and Cégielski, Heroult & Richard (2003) showed that the set of Erdős–Woods numbers is recursive.