Definition An Erdős–Woods number is a positive integer k > 1 such that there exists a sequence of k consecutive integers with the property that none of the numbers in the sequence is coprime to both endpoints of the sequence.
Overview Erdős–Woods numbers arise in number theory and are connected to questions concerning the distribution of prime factors in intervals of consecutive integers. The concept originates from a conjecture by Paul Erdős and was further explored by Alan R. Woods in his 1981 thesis. Woods investigated whether, for any interval of consecutive integers of a certain length, at least one element in the interval must be coprime to both endpoints. He initially conjectured this was always true, but later found counterexamples, leading to the identification of what are now called Erdős–Woods numbers.
The existence of such numbers implies that for certain lengths k, it is possible to find a sequence of k consecutive integers where every element shares a common factor greater than 1 with at least one of the endpoints.
Etymology/Origin The term "Erdős–Woods number" is derived from the names of mathematician Paul Erdős, a prolific contributor to number theory, and Alan R. Woods, who studied these sequences while working on his doctoral research at the University of Manchester. The concept stems from Woods’ investigation of coprimality in intervals of integers, inspired in part by questions posed by Erdős.
Characteristics
- The smallest Erdős–Woods number is 16. This means there exists a sequence of 16 consecutive integers (e.g., starting at 2184) such that each number in the interval shares a nontrivial common factor with at least one of the endpoints.
- Not all integers greater than 1 are Erdős–Woods numbers; their distribution is sparse.
- There are infinitely many Erdős–Woods numbers.
- The determination of whether a number k is an Erdős–Woods number involves checking for sequences of k consecutive integers where the coprimality condition fails for all interior elements relative to both endpoints.
Related Topics
- Number theory
- Coprimality
- Prime factorization
- Conjectures of Paul Erdős
- Interval arithmetic
- Diophantine approximation (indirectly)
- Model theory (Woods' original context involved logic and non-standard models of arithmetic)
Accurate information on the full classification and algorithmic generation of Erdős–Woods numbers remains an area of ongoing research, though computational methods have identified many such numbers.