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  Volume 7, Issue 4, Article 142
Inequalities Related to the Unitary Analogue of Lehmer Problem

    Authors: V. Siva Rama Prasad, Uma Dixit,  
    Keywords: Lehmer Problem, Unitary analogue of Lehmer problem.  
    Date Received: 15/05/06  
    Date Accepted: 20/06/06  
    Subject Codes:


    Editors: Jozsef Sandor,  

Observing that $ phi(n)$ divides $ n-1$ if $ n$ is a prime, where $ phi(n)$ is the well known Euler function, Lehmer has asked whether there is any composite number $ n$ with this property. For this unsolved problem, partial answers were given by several researchers. Considering the unitary analogue $ phi^ast(n)$ of $ phi(n)$, Subbarao noted that $ phi^ast(n)$ divides $ n-1$ , if $ n$ is the power of a prime; and sought for integers $ n$ other than prime powers which satisfy this condition. In this paper we improve two inequalities, established by Subbarao and Siva Rama Prasad [5], to be satisfied by $ n$ for $ phi^ast(n)$ which divides $ n-1$.

$ [5]$ M.V. Subbarao and V. Siva Rama Prasad, Some analogues of a Lehmer problem on the totient function, Rocky Mountain Journal of Mathematics; Vol. 15, Number 2: Spring 1985, 609-619.

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