  We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$
$set$, $\Lambda_P$, which comprises the irrationals whose continued fraction
entries are prime numbers. We prove that the Hausdorff measure of the prime
Cantor set cannot be finite and positive with respect to any sufficiently
regular dimension function, thus negatively answering a question of Mauldin
(2013) for this class of dimension functions. By contrast, under a reasonable
number-theoretic conjecture we prove that the packing measure of the conformal
measure on the prime Cantor set is in fact positive and finite with respect to
the dimension function $\psi(r) = r^\delta \log^{-2\delta}\log(1/r)$, where
$\delta$ is the dimension (conformal, Hausdorff, and packing) of the prime
Cantor set.
