On distribution properties of cubic residues

: In this paper, we use the elementary methods, the properties of the Gauss sums and the estimate for character sums to study the calculating problems of a certain cubic residues modulo p , and give some interesting identities and asymptotic formulas for their counting functions.


Introduction
Let p be an odd prime, and a be an integer with (a, p) = 1. In order to study of the properties of quadratic residues modulo p, Legendre first introduced the characteristic function of the quadratic residues a p , which was later called Legendre's symbol as follows: if a is a quadratic residue modulo p; −1, if a is a quadratic non-residue modulo p; 0 if p | a.
The introduction of this symbol greatly promotes the research about the properties of the quadratic residues and non-residues modulo p. Therefore, a great number of scholars began to study related work and obtained a series of valuable research results. For example, if p be an odd prime with p = 4k + 1, then for any quadratic residue r and quadratic non-residue s modulo p, one has the identity (see [2]: Theorem 4-11) where a −1 is the inverse of a mod p. In detail, a −1 satisfy the equation x · a ≡ 1 mod p.
Recently, Wang Tingting and Lv Xingxing [5] studied the distribution properties of a certain quadratic residues and non-residues modulo p, and proved the following two conclusions: Theorem A. For any prime p with p ≡ 3 mod 4, one has the identities where N(p, 1) denotes the number of all integers 1 ≤ a ≤ p − 1 such that a, a + a −1 and a − a −1 are all quadratic residues modulo p, N(p, −1) denotes the number of all integers 1 ≤ a ≤ p − 1 such that a is a quadratic non-residue modulo p, a + a −1 and a − a −1 are quadratic residues modulo p. Theorem B. For any prime p with p ≡ 1 mod 4, one has the asymptotic formulas where we have the estimates |E(p, 1)| ≤ 3 4 · √ p, |E 1 (p, 1)| ≤ 5 4 · √ p, |E(p, −1)| ≤ 3 4 · √ p and |E 1 (p, −1)| ≤ 5 4 · √ p.
As two corollaries of these results, Wang Tingting and Lv Xingxing [5] solved two open problems proposed by professor Sun Zhiwei. That is, they proved: For any prime p ≥ 101, there is at least one integer a, such that a, a+a −1 and a−a −1 are all quadratic residues modulo p.
For any prime p ≥ 18, there is at least one quadratic non-residue a mod p, such that a + a −1 and a − a −1 are quadratic residues modulo p.
In the field of the quadratic residues research, it is worth mentioning Z. H. Sun's important work [6]. We believe that Theorem A and B can also be derived from earlier results in [6]. All research works are very meaningful. In fact, the distribution of power residues plays an vital role in mathematics and cryptography, a number of important number theory and information security problems are closely related to it. A survey on this can be found in F. L. Ţ iplea, S. Iftene, G. Teşeleanu, A. M. Nica [10]. Therefore, it is necessary to continue to study the distribution properties of the power residues.
Inspired by the works in [5] and [6], we will naturally ask that what happens to the cubic residues modulo p?
It is clear that if (p − 1, 3) = 1, then for any integer a with (a, p) = 1, the congruence equation x 3 ≡ a mod p has one solution. So it is a trivial situation in the condition of (p − 1, 3) = 1. But if p ≡ 1 mod 3, then what happens? This paper focuses on such problems. Let p be a prime with p ≡ 1 mod 3, and N(p) denotes the number of all integers 1 < a < p − 1 such that a + a −1 and a − a −1 are cubic residues modulo p. In this paper, using the elementary methods, the properties of the third-order characters and Gauss sums, and the estimate for character sums, we are going to give an exact identity and some asymptotic formulas for N(p). Our main results are summarized as following three conclusions: Theorem 1. Let p be an odd prime with p ≡ 7 mod 12. If 2 is a cubic residue mod p, then we have the identity where d is uniquely determined by 4p = d 2 + 27b 2 and d ≡ 1 mod 3. Theorem 2. Let p be an odd prime with p ≡ 7 mod 12. If 2 is not a cubic residue mod p, then we have the asymptotic formula where we have the estimates |E(p)| ≤ 2 3 · √ p.
Theorem 3. Let p be an odd prime with p ≡ 1 mod 12, then we have the asymptotic formula where we have the estimates |E 1 (p)| < 26 9 · √ p.
According to our theorems, we can deduce the following: Corollary. Let p > 700 be a prime with p ≡ 1 mod 3, then there exists at least one integer 1 < a < p−1 such that a + a −1 and a − a −1 are cubic residues mod p.

Several lemmas
To prove our main results, we need following several basic lemmas. For simplicity, there is no need to repeat some elementary knowledge of number theory and analytic number theory, which can be found in references [1][2][3]. Lemma 1. Let p be a prime with p ≡ 1 mod 3. Then for any third-order character λ mod p, we have the identity where τ(χ) = Proof. Note that p ≡ 3 mod 4 and −1 p = −1, so based on the properties of the Legendre's symbol and complete residue system mod p, we obtain Similarly, we can also deduce that Combining (2.1) and (2.2) we have the identity This proves Lemma 2. Lemma 3. Let p be an odd prime with p ≡ 1 mod 3. Then for any third-order character λ mod p, we have the identity Proof. Write χ 2 = * p . Note that the identities λ = λ 2 , from the properties of the Gauss sums and the Legendre's symbol mod p we have Now according to the properties of complete residue system mod p, we have On the other hand, from the properties of Legendre's symbol mod p we also have Since λ 3 (2) = 1, so applying (2.4) and (2.5) we can deduce the identity Note that τ 2 (χ 2 ) = χ 2 (−1)p and τ(λ)τ λ = τ(λ)τ(λ) = p, combining Lemma 1, (2.3) and (2.6) we can deduce the identity This proves Lemma 3.

Lemma 4.
Let p be an odd prime with p ≡ 1 mod 3. Then for any third-order character λ mod p, we have the identity Proof. It can be deduced from the same methods of proving Lemma 3. Lemma 5. Let p be an odd prime with p ≡ 7 mod 12. Then for any third-order character λ mod p, we have the identity Proof. Note that the identity From the properties of the Legendre's symbol mod p we have Applying (2.7) and Lemma 3 we may immediately get This proves Lemma 5.

Conclusion
The main results of this paper are three theorems and a corollary. Theorem 1 obtained an exact formula for N(p) with p = 12k + 7 and 2 is a cubic residue modulo p. Theorem 2 and Theorem 3 established two asymptotic formulas for N(p) with p ≡ 1 mod 3 and 2 is not a cubic residue modulo p. At the same time, we also give two sharp upper bound estimates for the error terms. As some applications, we also deduced following corollary: If p > 700 is a prime with p ≡ 1 mod 3, then there is at least one integer 1 < a < p − 1 such that a + a −1 and a − a −1 are cubic residues modulo p.