Consecutive non-square non-primitive tuples in finite fields

Abstract

Let $q$ be an odd prime power and put \[ θ_q=\frac{φ(q-1)}{q-1}. \] 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 $\ell$th powers, where $\ell$ is an odd prime divisor of $q-1$. More precisely, if $k\ge 2$, $\text{char}{\mathbb{F}}_q\ge k$, and \[ q>(k-1)^2(2\ell)^{2k}, \] then ${\mathbb{F}}_q$ contains $k$ consecutive non-square $\ell$th powers. Combining this result with a finite computation, we prove that $\hat{I}{¸}_q<4/15$ guarantees the existence of three consecutive NSNP elements. On the boundary $\hat{I}{¸}_q=4/15$, the only exceptions are \[ \mathbb{F}_{31},\quad \mathbb{F}_{61},\quad \mathbb{F}_{121}. \] In particular, the constant $4/15$ is best possible.