Bounds for consecutive kth power residues in the Eisenstein integers

This paper extends to the Eisenstein integers a + b? (a, bτZ, ?² + ? + 1 = 0) the problem of the existence of a bound on the size of a sequence of m consecutive kth powe residues of p, for all but a finite number of primes p and independent of p. The least such bound is denoted by Λ_E(k, m). It is shown that Λ_E(k, 2) is finite for k = 2, 3, 4 or 6n + 1. On the other hand, for every k, Λ_E(2k, 3) = Λ_E(3k, 4) = Λ_E(k, 6) = ∞. Similar results are obtained for the related bound for m consecutives all in the same coset modulo the subgroup of kth power residues.