In this paper it is conjectured that for any positive integer $k$ there are infinitely many primes $p$ with the continued fraction expansion of $\sqrt{p}$ having length $k$ (Conjecture 5.1, https://web.williams.edu/Mathematics/sjmiller/public_html/mathlab/public_html/jr02fall/Periodicity/alexajp.pdf )
Do we have the weaker result that for any positive integer $k$ there exists at least one prime $p$ with the continued fraction expansion of $\sqrt{p}$ having length $k$ ?
Or perhaps relax the restriction of prime $p$ to any positive integer $n$ ?