It is thought that the totient valence function 
,
i.e., if there is an 
such that 
, then there are at least two solutions 
. This assertion is called Carmichael's totient function conjecture
and is equivalent to the statement that there exists an 
such that 
(Ribenboim 1996, pp. 39-40).
Dickson (2005, p. 137) states that the conjecture was proved by Carmichael (1907), who also developed a method of finding the solution (Carmichael 1909). The result
also appears as in exercise in Carmichael (1914). However, Carmichael (1922) subsequently
discovered an error in the proof, and the conjecture currently remains open. Any
counterexample to the conjecture must have more than 
digits (Schlafly and Wagon 1994;
conservatively given as 
in Conway and Guy 1996, p. 155). This result was extended
by Ford (1999), who showed that any counterexample must have more than 
digits.
Ford (1998ab) showed that if there is a counterexample to Carmichael's conjecture, then a positive proportion of totients are counterexamples.
Sierpiński's conjecture states that all integers 
appear as multiplicities of the totient
valence function.
-Function." Bull. Amer. Math. Soc. 13,
241-243, 1907.Carmichael, R. D. "Notes on the Simplex Theory
of Numbers." Bull. Amer. Math. Soc. 15, 217-223, 1909.Carmichael,
R. D. 
-Function." Bull. Amer. Math. Soc. 28,
109-110, 1922.Conway, J. H. and Guy, R. K. 
." Ann. Math. 150, 283-311, 1999.Guy,
R. K. "Carmichael's Conjecture." §B39 in 
."
Math. Comput. 63, 415-419, 1994.