1. Introduction
Let
p be an odd prime and
be an integer. For any integer
a with
, if the congruence equation
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
, then we call
a quadratic residue or quadratic non-residue modulo
p. Legendre first introduced the characteristic function of the quadratic residues
, which was later called Legendre’s symbol. It is defined as follows:
In particular, we have
,
and the quadratic reciprocity law
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 , there is at least one integer a, such that a, and are all quadratic residues modulo p.
(B). For any prime , there is at least one quadratic non-residue , such that and are quadratic residues modulo p, where is defined as .
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
, 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
.
Let p be a prime with , and denotes the number of all integers such that and 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 , 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 , then we have Theorem 2. Let p be an odd prime with , then we have the identitywhere we have the estimates . From our theorems, we can also deduce the following two corollaries:
Corollary 1. Let p be an odd prime with , and then we have the congruencewhich implies that Corollary 2. Let be a prime with , then there exists at least one integer a such that and 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.
2. Several Lemmas
In this section, we need to prove several simple lemmas. For ease of understanding, we first define the symbols that appear below:
denotes the classical Gauss sum
where
is an integer,
is any Dirichlet character
, and
.
denotes any fourth-order character , which is , .
denotes the Legendre’s symbol .
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
. 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 . Then, for any fourth-order character , one has the identitywhere denotes the Legendre’s symbol and . Proof. This result is Theorem 1 in Chen and Zhang [
21]. □
Lemma 2. Let p be an odd prime with . Then, for any fourth-order character , we have the identityand Proof. Note that
,
and
, so, from the properties of the Legendre’s symbol and complete residue system
, 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
,
,
, from Lemma 1, and the properties of Gauss sums, we have
or
This proves the second formula in Lemma 2. □
Lemma 3. Let p be an odd prime with . Then, we have the identity Proof. Note that
, from the method of proving Lemma 2, the properties of the Gauss sums and the Legendre’s symbol
we have
This proves Lemma 3. □
Lemma 4. Let p be an odd prime with . Then, we have the identity Proof. Note that
,
, and for any integer
a with
, we have
. Thus,
Similarly, we can also deduce the identity
From (3) and (4), we can deduce Lemma 4. □
Lemma 5. Let p be an odd prime with . Then, for any fourth-order character , we have the identities Proof. Note that
, 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. □
Lemma 6. Let p be an odd prime with . For any fourth-order character , if , then Proof. Since
,
and
, note that
or
, we have
Therefore, is a real number. However, , and we must have or .
If
, then we have
and
If
, then we have
and
This proves Lemma 6. □
3. 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
, there must exist an integer
such that
or
. Let
denote any fourth-order character modulo
p. Then, for any integer
n with
, we have the characteristic function:
Since
and
, from Lemma 6 and the definition of
, we have
If
, then note that
from (6), Lemma 2–Lemma 5, we have
This proves Theorem 1.
If
p is a prime with
, then note that
,
,
and
, from (6), we have
Since, in this case, we can not get those formulas like Lemma 3–Lemma 5. In this time, we only use the trivial estimates (see Weil’s work [
22] or Bourgain et al. [
23]):
Combining (7)–(16) we have asymptotic formula
where
satisfies the estimate
.
Now, we prove Corollary 1. Since
is an integer, from Theorem 1, we have
or
In order to prove Corollary 2, we just have to make sure that
. From Theorem 1 and Theorem 2, we just have the inequalities
and
Note that, from the estimate
we can get
for solving inequalities (17) and (18).
This completes the proofs of our all results.
4. Conclusions
The main results of this paper are two theorems and two corollaries. For prime
p with
, Theorem 1 of this paper gives an exact calculation formula for
, which contains an interesting constant
This
is closely related to prime
p. In fact, we have (see [
20]: Theorem 4–11)
where
s denotes any quadratic non-residue modulo
p.
We know very little about the arithmetic properties of , or even its parity. Obviously, for any prime p with , Corollary 1 gives for the first time a nontrivial congruence property of .
For any prime p with , we naturally ask the following two problems:
- (1).
Whether there exists an exact computing formula for ?
- (2).
What is the residue of ?
Interested readers are advised to study them with us.