A RATIO ASSOCIATED WITH 0(x) = n

Let $(x) be Euler's totient function. The literature on solving the equation cj)0) = n (see [1, pp. 221-223], [2-5], [6, pp. 50-55, problems B36-B42], [7-11], [12, pp. 228-256], and the references therein) can be viewed as a collection of open problems. For n = 2, we essentially have the problem of factoring the Fermat numbers. Another notorious example is Carmichaels conjecture [3, 7] that if a solution exists it is not unique. Some results (e.g., Example 15 of [12, pp. 238-239]) can be established on the basis of Schinzel's Conjecture H [12, p. 128] of which the twin prime conjecture is a very special case. See also [10, 11]. Here we define a new ratio R(ri) that is associated with this equation in a very natural way. Our main result, Theorem 3 of §3, is that R(ji) can be arbitrarily large. This can be read independently of §25 where the highest power of 2 dividing R(n) is studied. To define R(n) , let Ln be the least common multiple of all solutions of $(x) = n. Then, let Gn be the greatest common divisor of all numbers a 1, where a is in the reduced residue system modulo Ln given by