Erdős–Woods number

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 $(a, a + 1, \dots, a + k)$ 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 $1$ .

The first few Erdős–Woods numbers are

(Arguably 0 and 1 could also be included as trivial entries.)

Investigation of such numbers stemmed from the following prior conjecture by Paul Erdős:

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, $[2184, 2185, \dots, 2200]$ , 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.

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