On one of Murthy-Ashbacher's conjectures related to Euler's totient function
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A. Murthy and C. Ashbacher formulated a lot of open problems and conjectures in the area of elementary number theory. One of them is: For every positive integer k there exists a number n such that n '(n) = k: In the paper it is shown that this conjecture is not valid. It is proved that the only values of k for which the Diophantine equation k:'(n) = n is valid, are k = 1; 2; 3, where ' is Euler's totient function.
Abstract
A. Murthy and C. Ashbacher formulated a lot of open problems and conjectures in the area of elementary number theory. One of them is: For every positive integer k there exists a number n such that n '(n) = k: In the paper it is shown that this conjecture is not valid. It is proved that the only values of k for which the Diophantine equation k:'(n) = n is valid, are k = 1; 2; 3, where ' is Euler's totient function.