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Upper bound of period length of continued fraction representation of very composite number square root

Upper bound of period length of continued fraction representation of very composite number square root

EDIT: Dror answered this to my complete satisfaction. The simple continued fraction for $$ \sqrt{n^2 + 1}$$ has length exactly exactly the same, no dependence on $n$!!!!! No lower bound of the type I wanted is possible. The adjacent reduced indefinite binary quadratic forms in the complete cycle are $$ \langle 1, \; 2 n, \; -1 \rangle $$ $$ \langle -1, \; 2 n, \; 1 \rangle $$ $$ \langle 1, \; 2 n, \; -1 \rangle $$ for a cycle length of 2.

EDIT: evidently Schinzel studied these "sleepers, creepers, jeepers, etc." with known very short continued fraction lengths. So Dror had explicitly asked me whether I wanted to disregard those but I could not recognize the terminology and gave a misleading answer. So my method may be useless, but the revised question stands. And the person who knows how to correctly describe the question is Dror Speiser.

This question began with Dror Speiser's answer to

EDIT: Dror answered this to my complete satisfaction. The simple continued fraction for $$ \sqrt{n^2 + 1}$$ has length exactly the same, no dependence on $n$!!!!! No lower bound of the type I wanted is possible. The adjacent reduced indefinite binary quadratic forms in the complete cycle are $$ \langle 1, \; 2 n, \; -1 \rangle $$ $$ \langle -1, \; 2 n, \; 1 \rangle $$ $$ \langle 1, \; 2 n, \; -1 \rangle $$ for a cycle length of 2.

EDIT: evidently Schinzel studied these "sleepers, creepers, jeepers, etc." with known very short continued fraction lengths. So Dror had explicitly asked me whether I wanted to disregard those but I could not recognize the terminology and gave a misleading answer. So my method may be useless, but the revised question stands.

ORIGINAL QUESTION: This question began with Dror Speiser's answer to

This question began with Dror Speiser's answer to

EDIT: evidently Schinzel studied these "sleepers, creepers, jeepers, etc." with known very short continued fraction lengths. So Dror had explicitly asked me whether I wanted to disregard those but I could not recognize the terminology and gave a misleading answer. So my method may be useless, but the revised question stands. And the person who knows how to correctly describe the question is Dror Speiser.

This question began with Dror Speiser's answer to