Representation of the elements of the finite field \(\mathbb{F}_p\) by fractions (Q2338604)

The authors reword Thue's Lemma, see e.g. \textit{V. Shoup} [A computational introduction to number theory and algebra. 2nd ed. Cambridge: Cambridge University Press (2009; Zbl 1196.11002)] to state that for any odd prime \(p\), \[ (\mathbb{Z}/p \mathbb{Z})^* = \{\pm a/b \mid 1 \le a, b < \sqrt{p}\}.\] However, when \(p\) is replaced by \(n= 2p\) (and \(p>3\)), by considering the element \(p-2\) in \((\mathbb{Z}/p \mathbb{Z})^*\) they show that the bound of \(\sqrt{n}\) must here be replaced by at least the ceiling of \(\dfrac{p+3-\frac{p}{3}}{3}\), where \( \frac{p}{3}\) is the Legendre symbol, and give a conjecture for the exact bound in general.