Question 6
A real sequence $(a_n)$ is defined by
$$a_0=1,\qquad a_{n+1}=\frac12\left(a_n+\frac1{3a_n}\right)\quad\text{for every }n=1,2,3,\ldots.$$
Prove that, for every integer $n$, the number
$$A_n=\frac3{3a_n^2-1}$$
is a perfect square and has at least $n$ distinct prime divisors.
$$a_0=1,\qquad a_{n+1}=\frac12\left(a_n+\frac1{3a_n}\right)\quad\text{for every }n=1,2,3,\ldots.$$
Prove that, for every integer $n$, the number
$$A_n=\frac3{3a_n^2-1}$$
is a perfect square and has at least $n$ distinct prime divisors.