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Abstract.
The proof of the existence of infinitely many Carmichael numbers depends heavily upon the study of the Carmichael lambda function 
. In this paper, we study which types of forms this quantity can and cannot take. In particular, for a Carmichael number m, we prove that this 
can never
be of the form 
. Moreover, we prove that if this is of the form 
, then either 
or 127 and m is one of just eight possible values or else m is divisible by a Fermat prime that is not currently among the known Fermat primes.
Received: 2011-04-02
Revised: 2012-01-24
Accepted: 2012-04-05
Published Online: 2012-10-02
Published in Print: 2012-10-01
© 2012 by Walter de Gruyter Berlin Boston
