number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, 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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

of consecutive integers, each of the elements has a non-trivial

common factor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

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 divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

s or is greater than .

Examples

The first Erdős–Woods numbers are : 16, 22, 34, 36, 46, 56, 64, 66, 70, 76, 78, 86, 88, 92, 94, 96,

100 100 or one hundred ( Roman numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to de ...

, 106,

112 112 may refer to: *112 (number), the natural number following 111 and preceding 113 *112 (band), an American R&B quartet from Atlanta, Georgia **112 (album), ''112'' (album), album from the band of the same name *112 (emergency telephone number), t ...

, 116 … .

History

Investigation of such numbers stemmed from the following prior conjecture by Paul Erdős: :There exists a positive integer such that every integer is uniquely determined by the list of prime divisors of . Alan R. Woods investigated this question for his 1981 thesis. Woods conjectured that whenever , the interval always includes a number

coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...

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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of Erdős–Woods numbers is

recursive Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

References

External links

* {{DEFAULTSORT:Erdos-Woods number Woods number Integer sequences