A prime chain is a sequence p1, ..., pk
of primes for which pj | (pj+1-1)
for all j, e.g. 3, 7, 29, 59.
We study various problems about counting prime chains with certain properties,
in particular the
problem of counting chains with a given p1 and with
pk < x. This last problem has an application
to the distribution of the Carmichael function λ(n).
The Pratt tree for a prime p is defined recursi-
vely as the tree with root node p, and below p are links
to the Pratt trees for primes q dividing p-1
(the leaves are all labeled with 2). Our main questions center around
the depth of such trees. We
prove non-trivial upper and lower estimates on the depth, valid
for almost all p. We also examine
the behavior of a stochastic model of Pratt trees based on a random
fragmentation process, which
suggests (crudely) that the depth of the Pratt tree has normal order
e log log p.