摘要
设A是由n个互不相同的正整数ai组成的序列a1<a2<…<an,1970年,Graham猜测:maxi,jai(ai,aj)≥n.有许多数学家研究过这一猜想,直到1996年,Balasubramanian和Soundararajan完全解决了这一问题,但证明极其复杂.1999年,Granville和Roesler提出了一个有关两个正整数序列A和B的猜想:集合aagcd(a,b),gcd(ba,b),a∈A,b∈B中的最大元素≥min(|A|,|B|).当取A=B时,此猜想即为Graham猜想.本文证明了若序列A和B中至少都有一项是素数时,猜想成立.
In 1970, R. L. Graham asked if the following is true： Let A be a finite sequence of n different positive integers ai, 1≤ i ≤ n. Then maxi·j ai/（ai,aj）≥ n . Many number theorists have studied the problem. In 1996, this old conjecture though long and complicated, was proved correct in an outstandingly original by Balasubramanian and Soundararajan. Recently, Granville and Roesler conjectured that the largest element of the set {a/gcd（a,b）,b/gcd（a,b）,a∈A,6∈B}is ≥ min（ ｜ A ｜, ｜ B ｜ ）, which generalizes Graham＇s conjecture. In this paper the conjecture is verified when both A and B have a prime.
出处
《浙江大学学报（理学版）》
CAS
CSCD
北大核心
2006年第1期1-2,13,共3页
Journal of Zhejiang University（Science Edition）
基金
国家自然科学基金资助项目(10371107)
