Erdős–Woods number


In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property:
there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common factor with one of the endpoints. In other words, is an Erdős–Woods number if there exists a positive integer such that for each integer between and, at least one of the greatest common divisors or is greater than.

Examples

The first few Erdős–Woods numbers are

History

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, the interval always includes a number coprime to both endpoints. It was only later that he found the first counterexample,, with. The existence of this counterexample shows that 16 is an Erdős–Woods number.
proved that there are infinitely many Erdős–Woods numbers, and showed that the set of Erdős–Woods numbers is recursive.