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, …, 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.

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[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 (1989) proved that there are infinitely many Erdős–Woods numbers,[2] and Cégielski, Heroult & Richard (2003) showed that the set of Erdős–Woods numbers is recursive.[3]

References

  1. ↑ Alan L. Woods, Some problems in logic and number theory, and their connections. Ph.D. thesis, University of Manchester, 1981. Available online at http://school.maths.uwa.edu.au/~woods/thesis/WoodsPhDThesis.pdf (accessed July 2012)
  2. ↑ Dowe, David L. (1989), "On the existence of sequences of co-prime pairs of integers", J. Austral. Math. Soc. (A), 47: 84–89, doi:10.1017/S1446788700031220.
  3. ↑ Cégielski, Patrick; Heroult, François; Richard, Denis (2003), "On the amplitude of intervals of natural numbers whose every element has a common prime divisor with at least an extremity", Theoretical Computer Science, 303 (1): 53–62, doi:10.1016/S0304-3975(02)00444-9.

External links

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