It is a known fact that for a Prime $P$, $P\equiv 1$ mod $4$ iff the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd. I have an elementary proof of this using the classical result: $P\equiv 1$ mod $4$ iff $x^2-Py^2=-1$ has integer solutions and a proof that $x^2-Py^2=-1$ has integer solutions iff the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd.

I have tried repeatedly to give a direct elementary proof that $P\equiv 1$ mod $4$ implies that the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd, but cant seem to figure it out. (I have a direct elementary proof of the converse)

Does anyone know of an elementary proof of this result or where I may find one?