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We prove that (i) implies (ii). Since $b$ is primitive modulo $p$, writeBeing primitive modulo $p^2$ is equivalent to $p\nmid c$. The binomial theorem then gives inductivelyIt follows that the order modulo $p^m$ is exactlyso $b$ is primitive modulo every $p^m$. This is lifting a primitive root to odd prime powers. The implication (ii)$\Rightarrow$(i) is immediate by taking $m=2$.

By lifting a primitive root to odd prime powers, it is enough to test primitivity modulo $25$. The positive integers below eight either fail to be units or have orderswhereasTherefore the three smallest required integers are