On a conjecture of Carmichael
Victor L. Klee · Bulletin of the American Mathematical Society · 1947
V. L. KLEE, JR. 1 Carmichael [ l ] 2 conjectured that for no integer n can the equation (x)=n ( being Euler's totient) have exactly one solution. To support the conjecture, he showed that each n for which there is a unique solution must satisfy a restriction which implies w>10. In this note we prove the validity of restrictions considerably stronger than those of Carmichael, and raise the lower bound on n to 10. We shall denote by X the set of all integers x for which (y)=<l)(x) implies y = x. (If the conjecture is correct, X is empty, and the theorems stated are vacuously satisfied.)