  In this note, we study the problem of existence of sequences of consecutive
1's in the periodic part of the continued fractions expansions of square roots
of primes. We prove unconditionally that, for a given $N\gg 1$, there are at
least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction
expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the
periodic part. We also present results of our computations related to the
considered problem and some related problems, formulate several open questions
and conjectures and get some results under the assumption of Hypothesis H of
Schinzel.
