循环小数的奇妙性质

The Fascinating Properties of the Recurring Decimals

文 [1]中叙述了三个‘世界上数学家还没有解决的难题 ,即所谓世界难题 .’本文部分地讨论了第二个问题 ,发现在一定条件下 ,许多分数的循环节作为一个数都是 9的倍数 ,证明一类分母是合数的真分数也具有文

Three unsolved puzzles by mathematicians in mathematics were stated in the paper ,the second problem is that:we find the values of both fractions 17=0. 1 ·4285 7 · and 123=0. 0 ·43478260869565217391 3 ·  such that 142+857=999,04347826086+95652173913=99999999999. There are analogic fractions 117,119,...,1109,1113,...,1223,1229,..., and the proper fractions with the same denominators. The sums of corresponding digits in first half period and second half period always are 9.Why? How many such numbers are there? Is there a simple genaral formula for these numbers? This paper solves preceding question partily.