In number theory, a positive integer k is said to be an ErdĆsâWoods number if it has the following property: there exists a positive integer a such that in the sequence (a, a + 1, âŠ, 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 ErdĆsâWoods 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) or gcd(a + i, a + k) is greater than 1.
The first ErdĆsâWoods numbers are
Investigation of such numbers stemmed from the following prior conjecture by Paul ErdĆs:
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, âŠ, 2200], with k = 16. The existence of this counterexample shows that 16 is an ErdĆsâWoods number.
Dowe (1989) proved that there are infinitely many ErdĆsâWoods numbers, [2] and CĂ©gielski, Heroult & Richard (2003) showed that the set of ErdĆsâWoods numbers is recursive. [3]
In number theory, a positive integer k is said to be an ErdĆsâWoods number if it has the following property: there exists a positive integer a such that in the sequence (a, a + 1, âŠ, 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 ErdĆsâWoods 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) or gcd(a + i, a + k) is greater than 1.
The first ErdĆsâWoods numbers are
Investigation of such numbers stemmed from the following prior conjecture by Paul ErdĆs:
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, âŠ, 2200], with k = 16. The existence of this counterexample shows that 16 is an ErdĆsâWoods number.
Dowe (1989) proved that there are infinitely many ErdĆsâWoods numbers, [2] and CĂ©gielski, Heroult & Richard (2003) showed that the set of ErdĆsâWoods numbers is recursive. [3]