Let $q$ be a power of prime $p$ and $\zeta_p$ be the complex $p$ th root
 of unity. We denote by
$\mathbb{F}_q$ the finite field of $q$ elements and by $Tr$ the absolute trace function $\mathbb{F}_q\rightarrow \mathbb{F}_p$ . The **Kloosterman sum** at a point $a \in \mathbb{F}_q$ is defined by the equation

 $\hspace{3cm} K_{q}(a)=\displaystyle\sum_{x\in\mathbb{F}_q^*}\psi(x+ax^{-1})$

where $\psi:\mathbb{F}_q\rightarrow \mathbb{Q}(\zeta_p)$ is the canonical additive character of $\mathbb{F}_q$ defined by $\psi(x)=\zeta_p^{Tr(x)}.$

**FACT.** It is known that $K_{q}(a)\in\mathbb{Z}$ for $p=2,3$.

>**QUESTION.** Does there exist a prime $p\; (\ne 2,3$) such that $K_{q}(a)\in \mathbb{Z} ?$