Carmichael猜想的一个标注  

A remark on Carmichael’s conjecture

作　　者：王恒洲 史三英[1] WANG Hengzhou;SHI Sanying(School of Mathematics,Hefei University of Technology,Hefei 230601,China)

机构地区：[1]合肥工业大学数学学院

出　　处：《合肥工业大学学报（自然科学版）》2020年第2期285-288,共4页Journal of Hefei University of Technology：Natural Science

基　　金：国家自然科学基金资助项目(11201107);安徽省自然科学基金资助项目(1208085QA01)

摘　　要：Carmichael猜想是数论的一个经典猜想,很多数学家都对该猜想做过各种研究。Carl Pomerance是第1个对该猜想进行理论研究的,并提出了该猜想的一个充分条件。通过引进φ-子集概念,文章研究了关于未知数n的方程φ(n)=x的解的个数,证明该猜想的一个充要条件为:Carmichael猜想成立当且仅当猜想在集{2^43^37^243k,k为任意正整数}上成立。The Carmichael’s conjecture is a classical conjecture of number theory. Klee, Masai and other mathematicians have done a variety of studies on this conjecture, but their work is limited to the studies of computational number theory. Carl Pomerance was the first to study this conjecture theoretically, and proved a sufficient condition for this conjecture. In this paper, by studying the number of solutions n to the equation φ(n)=x, we provide new research ideas for Carmichael’s conjecture. Finally, we find a necessary and sufficient condition of this conjecture, that is, Carmichael’s conjecture is true if and only if it is true in the set {2^43^37^243k,k∈N}.

关 键 词：Carmichael猜想 欧拉函数 φ-子集 素数 素因数集 

分 类 号：O156.1[理学—数学]