Bounds for consecutive \(k\)-th power residues in the Eisenstein integers (Q1215536)

Let \(k>1\), \(m >1\) be given integers. It is known that for all but a finite set of (exceptional) primes \(p\), there exists \(r\) such that \(r, r+1,\ldots, r+ m - 1\) are all \(k\)-th power residues modulo \(p\). Let \(r(k,m,p)\) be the least such \(r\), and let \(A(k,m) = \max r(k,m,p)\), taken over the non-exceptional primes. If the \(r\) are unbounded we write \(A(k,m) = \infty\). Many values of \(A(k,m)\) have been obtained, and the problem has been extended to primes in the Gaussian field \(K(i)\) by \textit{J. H. Jordan} [J. Number Theory 1, 477--485 (1969; Zbl 0186.08001)]. In the present paper, the problem is extended, in a similar way, to primes in the Eisenstein field \(K\left(e^{2\pi i/3}\right)\).