Small norm residues from square-root continued fractions ($|p_k^2-N{q}_k^2|<2\sqrt{N}$)

For nonsquare $N>1$, write the complete quotients after the initial term as $(\sqrt{N}+m)/d$. They exceed one, while their algebraic conjugates lie between minus one and zero. Thus $d<\sqrt{N}+m<2\sqrt{N}$. Comparing coefficients in $\sqrt{N}=(p_k{\alpha }_{k+1}+p_{k-1})/(q_k{\alpha }_{k+1}+q_{k-1})$ gives $p_k^2-N{q}_k^2=d_{k+1}(p_k{q}_{k-1}-q_k{p}_{k-1})$, and the last determinant has absolute value one. These small residues are useful candidates for smooth numbers in congruence-of-squares factorization.