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Periodic integer continued fractions (PICFs) are generalization of theregular periodic continued fractions (RPCFs). It is classical that a RPCFexpansion of an irrational number is unique. However, it is no longer uniquefor a PICF expansion. Hence it is a natural problem to determine all PICFexpansions of irrational numbers. In this paper, we determine certain type PICFexpansions of square roots of positive square-free integers. To obtain thisresult, it plays an important role to determine integer points on certain PCFvarieties appeared in Brock-Elkies-Jordan. As an application of these results,we obtain fundamental solutions of the Pell equations from PICF expansions ofsquare roots of positive square-free integers as well as the RPCF expansions.

Some periodic integer continued fraction expansions of $\sqrt{m}$ and application to the Pell equations