Consecutive non-square non-primitive tuples in finite fields

Let $q$ be an odd prime power and put $${\theta }_q=\frac{\phi (q-1)}{q-1}\mathrm{.}$$ Let ${\mathbb{F}}_q$ denote a finite field with $q$ elements, an element of ${\mathbb{F}}_q$ is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. We first obtain a general existence theorem for consecutive tuples of non-square $\mathrm{\ell }$th powers, where $\mathrm{\ell }$ is an odd prime divisor of $q-1$. More precisely, if $k\ge 2$, $\text{char}{\mathbb{F}}_q\ge k$, and $$q>(k-1{)}^2(2\mathrm{\ell }{)}^{2k},$$ then ${\mathbb{F}}_q$ contains $k$ consecutive non-square $\mathrm{\ell }$th powers. Combining this result with a finite computation, we prove that ${\theta }_q<4\mathrm{/}15$ guarantees the existence of three consecutive NSNP elements. On the boundary ${\theta }_q=4\mathrm{/}15$, the only exceptions are $${\mathbb{F}}_{31},{\mathbb{F}}_{61},{\mathbb{F}}_{121}\mathrm{.}$$ In particular, the constant $4\mathrm{/}15$ is best possible.