On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

Abstract EN

For any fixed integer k≥2 and integer r with r, p=1, it is clear that there exist k integers 1≤ai≤p-1 i=1, 2, …, k such that a1a2⋯ak≡r mod p.

Let N(k,r;p) denote the number of all a1, a2, ⋯ak such that a1a2⋯ak≡r mod p and 2†a1+a2+⋯ + ak.

In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties of N(k,r;p) and give two interesting asymptotic formulae for it.