On two questions of Ono. (Q1565793)

The authors provide an elementary proof of a well-known result: Theorem 1: If \((T,U)\) is the fundamental solution of \(x^2-py^2=1\), where \(p\) is prime, then \(T\equiv 1\pmod p\) if and only if \(p=2\) or \(p \equiv 7\pmod 8\). They also provide a variant of Lagrange's theorem on the solvability of \(x^2-py^2=\pm 1\) using continued fractions.