nt.number theory - Periods of Continued Fractions

Let $p=2$, $q=5$, so $pq=10$. Then $ell(p)=ell(q)=ell(pq)=1$, so if $c$ exists it must be 1. Now let $p=2$,
$q=17$, so $pq=34$. Then $ell(p)=ell(q)=1neell(pq)$, so if $c$ exists it must not be 1. Contradiction.

EDIT: Here's a little table which, I think, shows how unlikely it is that there's any simple equation relating the period lengths of the square roots of primes $p$ and $q$ and their product.
$$
matrix{p&q&pq&ell(p)&ell(q)&ell(pq)cr2&5&10&1&1&1cr2&17&34&1&1&4cr2&13&26&1&5&1cr2&7&14&1&4&4cr13&89&1157&5&5&1cr}
$$

So, if $ell(p)$ and $ell(q)$ are both 1 then $ell(pq)$ may or may not be 1; if $ell(p)$ and $ell(pq)$ are both 1 then $ell(q)$ may or may not be 1; if $ell(p)$ is 1 and $ell(pq)$ is not 1 then $ell(q)$ may or may not be 1; if $ell(pq)$ is 1 and $ell(q)$ is not 1 then $ell(p)$ may or may not be 1.