On the Quartic Residues and Their New Distribution Properties

: In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p , and give an exact calculation formula and asymptotic formula for their counting functions.


Introduction
Let p be an odd prime and k ≥ 2 be an integer. For any integer a with (a, p) = 1, if the congruence equation x k ≡ a mod p has solution, then we call a is a k-th residue modulo p. Otherwise, a is called a k-th non-residue modulo p. If k = 2, then we call a quadratic residue or quadratic non-residue modulo p. Legendre first introduced the characteristic function of the quadratic residues a p , which was later called Legendre's symbol. It is defined as follows: if a is a quadratic residue modulo p; −1, if a is a quadratic non-residue modulo p; 0, if p | a.
In particular, we have where p and q are two different odd primes. The study of k-th residues modulo p is one of the important pieces of content in elementary number theory and analytic number theory, and many number theory problems are closely related to them. Because of this, many scholars have been engaged in the research work in this field, and have made rich research results. It is worth mentioning that Sun Zhihong [1][2][3][4][5] has done a lot of profound research on the quartic residues; these bring us to the study of the distribution properties of various k-th residues modulo p. Some other papers related to quadratic residues and cubic residues modulo p can be found in references [6][7][8][9][10][11][12][13][14][15][16][17][18]. For example, recently, Wang Tingting and Lv Xingxing [6] studied the distribution properties of some special quadratic residues and non-residues modulo p, who obtained an exact calculation formula and a sharp asymptotic formula for its counting function.
As some applications, they solved two problems proposed by Sun Zhiwei. That is, they proved the following two interesting results: (A). For any prime p ≥ 101, there is at least one integer a, such that a, a + a and a − a are all quadratic residues modulo p.
(B). For any prime p ≥ 18, there is at least one quadratic non-residue a mod p, such that a + a and a − a are quadratic residues modulo p, where a is defined as aa ≡ 1 mod p.
As an extension of Wang Tingting and Lv Xingxing's work [6], a natural problem is the quartic residues modulo p. It is clear that, if p ≡ 3 mod 4, then the quadratic residue is the same as the quartic residue modulo p. In this time, the problem is trivial. Thus, we just consider the non-trivial case p ≡ 1 mod 4.
Let p be a prime with p ≡ 1 mod 4, and N(p) denotes the number of all integers 1 < a < p − 1 such that a + a and a − a are quartic residues modulo p.
In this paper, we will use the analytic methods, the properties of the classical Gauss sums, and the estimate for character sums to study the computational problems of N(p), and give an exact calculation formula and asymptotic formula for it. That is, we will prove the following two results: Theorem 1. Let p be an odd prime with p ≡ 5 mod 8, then we have Theorem 2. Let p be an odd prime with p ≡ 1 mod 8, then we have the identity where we have the estimates |E(p)| ≤ 15 4 · √ p.
From our theorems, we can also deduce the following two corollaries: Corollary 1. Let p be an odd prime with p ≡ 5 mod 8, and then we have the congruence

Corollary 2.
Let p > 3700 be a prime with p ≡ 1 mod 4, then there exists at least one integer a such that a + a and a − a are quartic residues modulo p.

Some notes:
Prior work gave us great inspiration for the research of this paper, but the methods we used is completely different from the methods in [6] or [1][2][3][4][5], where they all use elementary methods, so they can only get some qualitative results or asymptotic formulas. We used some analytic methods and the properties of the classical Gauss sums. Thus, an accurate calculation formula is obtained.
In addition, Corollary 2 is a very rough estimate deduced directly from our theorems. If we use some mathematical software, then the constant 3700 in Corollary 2 can be made much smaller.

Several Lemmas
In this section, we need to prove several simple lemmas. For ease of understanding, we first define the symbols that appear below: τ (χ) denotes the classical Gauss sum where q ≥ 3 is an integer, χ is any Dirichlet character modq, and e(y) = e 2πiy . λ denotes any fourth-order character mod p, which is λ = χ 0 , λ 4 = χ 0 . χ 2 = * p denotes the Legendre's symbol mod p. The basic knowledge required in this section can also be found in references [19,20]. We will decompose the proofs of our theorems into the following several lemmas by means of the characteristic function of the fourth-order character λ mod p. In the end, we only deal with some estimate for a certain character sums or calculations for some special Gauss sums. First, we have: Lemma 1. Let p be a prime with p ≡ 1 mod 4. Then, for any fourth-order character λ mod p, one has the identity where * p denotes the Legendre's symbol mod p and aa ≡ 1 mod p.
Proof. This result is Theorem 1 in Chen and Zhang [21].

Lemma 2.
Let p be an odd prime with p ≡ 5 mod 8. Then, for any fourth-order character λ mod p, we have the identity Proof. Note that p ≡ 5 mod 8, χ 2 = λ 2 = λ 2 and λ(−1) = −1, so, from the properties of the Legendre's symbol and complete residue system mod p, we have Similarly, we can also deduce that It is clear that the first formula in Lemma 2 follows from (1) and (2). Now, we prove the second one. Note that τ( from Lemma 1, and the properties of Gauss sums, we have This proves the second formula in Lemma 2.

Lemma 5.
Let p be an odd prime with p ≡ 5 mod 8. Then, for any fourth-order character λ mod p, we have the identities Proof. Note that λ(−1) = −1, from the properties of the reduced residue system modulo p, we have the identity which implies that From (5), we may immediately get This proves Lemma 5.

Proofs of the Theorems
In this section, we shall complete the proofs of our main results. First, we prove Theorem 1. For any prime p with p ≡ 5 mod 8, there must exist an integer 1 < r < p − 1 such that r 2 ≡ −1 mod p or r + r ≡ 0 mod p. Let λ denote any fourth-order character modulo p. Then, for any integer n with (n, p) = 1, we have the characteristic function: if n is a quartic residue modulo p; 0, if n is not a quartic residue modulo p.