Claims About a Periodic Recurrence

A sequence $\{a_n\}$ is periodic with period $T$ if there is a positive integer $T$ with $a_{n+T}=a_n$ for all $n$. Let $\{a_n\}$ satisfy $a_1=m$ ($m>0$) and

Determine, with proof, which of the following claims are false.

1. If $a_3=4$, then $m$ can take $3$ different values. 2. If $m=\sqrt 2$, then $\{a_n\}$ has period $3$. 3. For every integer $T\geqslant 2$, there exists $m>1$ making $\{a_n\}$ periodic with period $T$. 4. There exists a rational $m\geqslant 2$ making $\{a_n\}$ periodic.