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In number theory, a positive integerk 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 divisorsgcd(a, a + i) or gcd(a + i, a + k) is greater than 1.
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[1] 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, David L. (1989), "On the existence of sequences of co-prime pairs of integers", J. Austral. Math. Soc. Ser. A, 47: 84–89, doi:10.1017/S1446788700031220.
This page was last edited on 9 June 2024, at 07:49