A conjecture of Krishnamurthy on decimal periods and some allied problems
A conjecture of Krishnamurthy on decimal periods and some allied problems
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克里希那穆西关于十进制周期的猜想及一些相关问题
DOI：
10.1016/0022-314x(81)90016-0
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发表时间：
1981
影响因子：
0.7
通讯作者：
R. Odoni
中科院分区：
文献类型：
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作者：
R. Odoni
在[I]中，克里希那穆尔西证明，对于渐近三分之一的素数p> 5，I/p的小数周期是奇数，给出了支持他的猜想的数值证据。耶茨[2]提供了进一步的数值数据，而桑克斯[3]给出了一个启发式的论点，至少表明了克里希那穆尔西猜想的真实性。在本文中，我们给出了一个严格的证明猜想，在一个更一般的形式，也考虑了倒数的合数，而不仅仅是素数的周期。设g> 1为自然数。如果a和B是自然数，其中1< a< B且（a，B）=（a，g）=（B，g）= 1，则一个众所周知的基本结果是，分数u/B在按g的尺度展开时是周期性的，即其g-adic数的序列d，由a/B= fd，g-"，0< 4，< g，It= 1给出
In [I], Krishnamurthy conjectured that, for asymptotically one-third of primes p> 5, the decimal period of I/p is odd, giving numerical evidence in support of his conjecture. Further numerical data were supplied by Yates [2], while Shanks [3] gave a heuristic argument which at least suggested the truth of Krishnamurthy’s conjecture. In this paper we give a rigorous proof of the conjecture, in a rather more general form, and also consider the periods of reciprocals of composite numbers, not merely of primes. Let g> 1 be a natural number. If a and b are natural numbers with 1< a< b and (a, b)=(a, g)=(b, g)= 1, then it is a well-known, elementary result that the fraction u/b, when expanded g-adically (ie,“in the scale of g”) is periodic, ie, the sequence d, of its g-adic digits, given by a/b= fd, g-“, 0< 4,< g, It= 1