(i) State the law of quadratic reciprocity.

Prove that 5 is a quadratic residue modulo primes $p \equiv \pm 1 \quad(\bmod 10)$ and a quadratic non-residue modulo primes $p \equiv \pm 3 \quad(\bmod 10)$.

Determine whether 5 is a quadratic residue or non-residue modulo 119 and modulo $539 .$

(ii) Explain what is meant by the continued fraction of a real number $\theta$. Define the convergents to $\theta$ and write down the recurrence relations satisfied by their numerators and denominators.

Use the continued fraction method to find two solutions in positive integers $x, y$ of the equation $x^{2}-15 y^{2}=1$.