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The keyword here is Singer cycle. -- A Singer cycle in ${\rm GL}(n,q)$ is an element of order $q^n-1$. This is the largest possible order of an element of ${\rm GL}(n,q)$. As you are looking for element orders in ${\rm PSL}(n,q)$, you need to divide that order by the product of $q-1$ and the size of the centre of ${\rm GL}(n,q)$ (i.e. $\gcd(n,q-1)$) to obtain the value you are looking for.