Carmichael’s Conjecture on Preimages of Euler’s Totient Function

Establish that for every integer m > 1, the equation φ(n) = m has at least two distinct integer solutions n; equivalently, prove that the multiplicity A(m) = #{n ∈ N : φ(n) = m} is never equal to 1 for any m > 1.

Background

The paper briefly notes longstanding open problems related to Euler’s totient function φ, citing Carmichael’s Conjecture as one of them. Carmichael’s Conjecture asserts that the preimage size under φ is never exactly one for any value m > 1. This problem concerns the structure and distribution of values taken by φ and the cardinality of their preimage sets.

In the context of iterated totient sums studied in the paper, understanding the preimage multiplicities of φ is foundational: many structural results about φ-values under iteration rely on the arithmetic properties and frequency of φ-values, which are constrained if Carmichael’s Conjecture holds.