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LEHMER’S TOTIENT PROBLEM AND CARMICHAEL NUMBERS IN A PID

Lehmer’s totient problem consists of determining the set of positive integers n such that φ(n)|n−1 where φ is Euler’s totient function. It is not obvious whether there are any composite n satisfying this divisibility condition; in fact, any such composite n is a Carmichael number (although every known Carmichael number doesn’t actually have this property). We will generalize the above divisibility condition (with the cardinality [when finite] of the group of units in a quotient ring playing the role of φ(n)), construct a reasonable notion of Carmichael numbers in a PID and use a pair of handy short exact sequences to show how similar statements to those above follow in more generality. Also, we’ll pick up a couple of generalizations for classical identities involving φ along the way. Included will be a generalization of the work of Korselt and an extension of the work of Alford, Granville and Pomerance.