Consecutive units

Abstract:

Let $p$ be a prime $> 3$ , and let $\zeta$ be a primitive $p$th root of unity. Let $k$ be the maximum number of consecutive units of the cyclotomic field ${\mathbf {Q}}\left ( \zeta \right )$. It is shown that $k \leq \max \left ( {4,R,N} \right )$, where $R$ is the maximum number of consecutive residues modulo $p$ , and $N$ the maximum number of consecutive non-residues modulo $p$. This result implies that, for the primes $p > 3$ under 100,$k$ is exactly 4 for $p = 5,7,11,13,17,19,23,29,31,37,47,73$ (and possibly for the other primes as well). Another consequence is that $k < 2{p^{1/2}}$.