On the Golomb’s conjecture and Lehmer’s numbers

Abstract Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.


Introduction
Let p be an odd number. For each integer a with 1 Ä a Ä p 1, it is clear that there exists one and only one a with 1 Ä a Ä p 1 such that a a Á 1 mod p. Let M.p/ denote the set of cases in which a and a are of opposite parity. For convenience, we call such a number a with .2; a C a/ D 1 as a D. H. Lehmer's number. Professor D. H. Lehmer [1] asked us to study the properties of jM.p/j or at least to say something nontrivial about it, where jM.p/j denotes the number of all elements in M.p/. It is known that jM.p/j Á 2 or 0 mod 4 when p Á˙1 mod 4. Some related works can be found in references [2][3][4][5]. For example, W. Zhang [2] proved that for any odd q > 3, one has the asymptotic formula jM.q/j D 1 2 .q/ C O q 1 2 d 2 .q/ ln 2 q Á ; where .q/ is Euler function, and d.q/ is Dirichlet divisor function. Now we let N.p/ denote the set of all primitive roots a mod p with 1 Ä a Ä p 1 in which a and a are of opposite parity. For any integer d with 1 Ä d Ä p 1, Golomb [6] posed some problems related to primitive roots, one of which is that whether there exist two primitive roots a; b mod p such that a C b Á d mod p. Some related works can also be found in [7][8][9].
In this paper, we consider the following problem involving Golomb's conjecture and Lehmer's numbers: Do there exist two primitive roots a and b 2 N.p/ such that Regarding this problem, it seems that it has not been studied yet, or at least we have not seen any related result before. The problem is difficult and important, because it has close relations with the Golomb's conjecture and the Lehmer's numbers.
The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence equation a C b Á 1 mod p with a; b 2 N.p/, and give a sharper asymptotic formula for (1). That is, we shall prove the following: Theorem. Let p be an odd prime and S.p/ denote the number of all solutions of (1). Then we have the asymptotic formula where .n/ denotes the Euler function, and !.n/ denotes the number of all distinct prime divisors of n.

Several lemmas
In this section, we shall prove several lemmas, which will be used in the proof of our theorem. Hereinafter, we shall use many properties of Gauss sums and the hybrid exponential sums, all of these can be found in references [10][11], so they will not be repeated here. First we have the following: Lemma 2.1. Let p be an odd prime. Then for any integer c with .c; p/ D 1, we have the identity where ind c denotes the index of c relative to some fixed primitive root of p, .n/ is the M R obius function.
Lemma 2.2. Let p be an odd prime, R 1 .x/ and R 2 .x/ are two rational functions mod p, be any a non-principal character modp, and at least one of the conditions 1 Combining (2) and (3) If both 1 and 2 are not principal characters mod p, then from the properties of Gauss sums we have For any character mod p, note that the estimate j . /j Ä p p, from (5), (6) and (7) we may immediately deduce Lemma 2.5.

Proof of the theorem
In this section, we will use the lemmas in section two to complete the proof of our theorem. Note that if a C b Á 1 mod p, then a C b D p C 1 and b D p C 1 a D p a 1.