|
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. The first few Erdős–Woods numbers are :16, 22, 34, 36, 46, 56, 64, 66, 70 … . (Arguably 0 and 1 could also be included as trivial entries.) 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〔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)〕 that whenever ''k'' > 1, the interval (''a'' + ''k'' ) always includes a number coprime to both endpoints. It was only later that he found the first counterexample, (2185, …, 2200 ), with ''k'' = 16. proved that there are infinitely many Erdős–Woods numbers, and showed that the set of Erdős–Woods numbers is recursive. ==References== * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Erdős–Woods number」の詳細全文を読む スポンサード リンク
|