关于Graham猜想

On a Conjecture of Graham

本文证明了如下结果:若在正整数集N上定义二元运算a■b=a(a,b)^(-1),则<N;■,1>是一可换BCK一代数,其中(a,b)表示a与b的最大公因子。在此基础上还推广了前人关于Graham R L于1970年提出的一个猜想中的3个结果,证明了在下面任一情形下,Graham猜想都成立。即由a_1,a_2,…,a_n生成的<N;■,1>的子代数是关联的;A中含一个P_-数P的幂P~*,且对A中任二互异的数a_i与a_j有a_i≠a_j P^k,其中k为任一正整数。

In this paper, Let N be the set of all positive integers.A binary operation over N is defined by a b=a(a, b)^(-1)where (a, b) is the greatest commom divisor of a and b, Thcn<N; 1 > is a commutative BCK-algebra in which 1 is the natural number 1.Based on this conclusion, we generalized three results about the conjecture of Graham which have been proved by the former mathematicians and proved that the conjecture of Graham holds for three cases below:The subalgcbra of < N; 1 > generalized by a_1, a_2, …, a_is implicativc BCK-algcbra; A contains a P_-~* number; A contatins the power P^a of P_-~* number p, and a_i≠a_jp^k for any a_i and a_j in A where kis a any positive integers. Keywords:BCK-algcbra; P_-~* number; Positive integers.