Algebraic integers with continued fraction expansions containing palindromes and square roots with prescribed periods

We present a characterization of the algebraic integers with continued fraction expansions of the form [a0,a1,…,an,s¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯], where (a1,…,an) is a palindrome and s∈N≥1. In particular, we focus on the special case where (a1,…,an)=(m,…,m), providing a detailed characterizations of the corresponding algebraic integers and s in terms of Fibonacci polynomials. Then, we derive new expansions of square roots of integers with these periods, given m and n. Moreover, we explicitly determine the fundamental solutions of both positive and negative Pell's equations corresponding to this family of integers.