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For n ≥ 3 we obtain an improved estimate for the generalized Heilbronn sum ∑p−1 x=1 epn (yx pn−1 ), and use it to show that any interval I of points in Zpn of length |I| p1.825 for n = 2, |I| p2.959 for n = 3, |I| ≥ pn−3.269(34/151)n+o(1) for n ≥ 4, contains a (p− 1)-th root of unity. As a consequence, we derive an improved estimate for the Lind-Lehmer constant for the abelian group Zp , and improved estimates for Fermat quotients.

(p - 1)th Roots of unity mod pn, generalized Heilbronn sums, Lind-Lehmer constants, and Fermat quotients