Let F= be a rank two free group. A word W(a,b) in F is {\sl
primitive} if it, along with another group element, generates the group. It is
a {\sl palindrome} (with respect to a and b) if it reads the same forwards
and backwards. It is known that in a rank two free group any primitive element
is conjugate either to a palindrome or to the product of two palindromes, but
known iteration schemes for all primitive words give only a representative for
the conjugacy class. Here we derive a new iteration scheme that gives either
the unique palindrome in the conjugacy class or expresses the word as a unique
product of two unique palindromes. We denote these words by Ep/q where
p/q is rational number expressed in lowest terms. We prove that Ep/q is
a palindrome if pq is even and the unique product of two unique palindromes
if pq is odd. We prove that the pairs (Ep/q,Er/s) generate the group
when ∣ps−rq∣=1. This improves the previously known result that held only for
pq and rs both even. The derivation of the enumeration scheme also gives a
new proof of the known results about primitives.Comment: Final revisions, to appear J Algebr