A note on the primitive roots and the Golomb conjecture (Q2666163)

Summary: In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let \(p\) be a prime large enough. Then, for any positive integer \(n\) with \(p^{(1/2) + \varepsilon} \leq n < p\), there must exist two primitive roots \(\alpha\) and \(\beta\) modulo \(p\) with \(1<\alpha, \beta\leq n-1\) such that the equation \(n=\alpha+\beta\) holds, where \(0<\varepsilon<(1/2)\) is a fixed positive number. In other words, \(n\) can be expressed as the exact sum of two primitive roots modulo \(p\).