Abstract

We provide elementary proofs of the classical quadratic-residue criteria for -1, 2, 3, and 5 modulo odd primes. In addition, we determine the quadratic character of the integers a and b in the prime representations p=a²+b² and p=a²+2b². These results further illustrate the relationship between quadratic residues and binary quadratic forms.

1. Introduction

The quadratic character of the integers , , , and modulo odd primes constitute classical topics in elementary number theory. These criteria are traditionally derived from the Law of Quadratic Reciprocity together with its supplementary laws and occupy a central place in the theory of quadratic residues. The corresponding results and their various proofs can be found in many classical and modern references on number theory.

Classical number theory provides explicit criteria for determining whether , , , and are quadratic residues or quadratic non-residues modulo primes. Although these criteria are well known, their proofs can be approached from different perspectives, leading to a variety of techniques and arguments.

Historically, these results emerged gradually from the development of the theory of quadratic residues in the works of Euler, Legendre, and Gauss. Euler first studied special cases of quadratic residues in the 18th century, particularly in the period 1720–1750, identifying several concrete instances of quadratic behavior modulo primes. Later, in the late 18th century (around 1785), Legendre introduced a systematic treatment of quadratic residues and the notation of the Legendre symbol notation, laying the foundation for a more structured theory. The complete and unified formulation of these criteria was achieved by Gauss in his Disquisitiones Arithmeticae (1801), where the Law of Quadratic Reciprocity was proved and earlier partial results were fully generalized and unified.

The purpose of this paper is to present alternative elementary proofs of these quadratic-residue criteria. Our approach relies on elementary techniques from number theory. In this way, we provide alternative derivations of several classical results while remaining within an elementary framework.

Furthermore, we investigate the quadratic character of the coefficients appearing in the classical representations of prime numbers by binary quadratic forms. More precisely, for primes admitting representations of the form

and

we determine the quadratic character of the integers and modulo . These results further illustrate the close relationship between quadratic residues and the representation of primes by quadratic forms.

As is well known, for an odd prime , there are exactly quadratic residues and quadratic non-residues modulo . We denote by

the set of quadratic residues, and by

the set of quadratic non-residues modulo .

We recall the following well-known results in elementary number theory, which will be used throughout the paper. These results admit elementary proofs in the literature, without the use of quadratic characters or the law of quadratic reciprocity.

Theorem 1.1. ¹ The product of two quadratic residues, or of two quadratic non-residues, modulo , is a quadratic residue, whereas the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue.

Theorem 1.2. ² An odd prime is expressible as a sum of two squares if and only if .

Theorem 1.3. ³ A positive integer is properly representable as a sum of two squares if and only if the prime factors of are all of the form , except for the prime , which may occurs to at most the first power.

Theorem 1.4. ⁴ For an odd prime , we have:

(a)

(b)

(c)

We conclude this section by presenting the following two theorems, which will be used in Section 3.

Theorem 1.5. ⁵ Reciprocity law for Jacobi symbols. If and are positive odd integers with , then

Theorem 1.6. ⁶ (-th power Residue Criterion) Let be a prime and let be coprime to . Then is a -th power residue modulo if and only if

where .

2. Main Results

In this section, we present elementary proofs for determining the quadratic character of the integers , , and modulo primes. These results are classical in nature, but our approach provides alternative derivations based on elementary techniques from number theory, avoiding the use of the Law of Quadratic Reciprocity.

Theorem 2.1. If is an odd prime, then

Proof. The proof is divided into two cases.

Case 1. .

By Theorem 1.2, the prime can be written as

where . Then we have

Since and by Theorem 1.1. the product of two quadratic residues is a quadratic residue, it follows that

Case 2. .

From Theorem 1.3 it follows that the congruence

has no solution. Therefore, for every integer

Hence, it is clear that

Theorem 2.2. If is an odd prime, then

Proof. The proof is divided into four cases.

Case 1. .

By Theorem 2.1, the number is a quadratic residue modulo . Now, by Theorem 1.4 (a), the prime can be written as

where . Then we have

Since , and by Theorem 1.1. the product of two quadratic residues is a quadratic residue modulo , it follows that

Case 2. .

By Theorem 2.1, the integer is a quadratic non-residue modulo . By Theorem 1.4. (a), we have , . From congruence

since , , and taking into account that the product of a quadratic residues with a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 3. .

From Theorem 1.4. (a), it follows that the congruence

has no solution. Then, for every , we obtain

Thus, we have . Since , and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 4. .

In the same way as in the case 3, we prove that . Since , and the product of two quadratic non-residue is a quadratic residue modulo , it follows that

Theorem 2.3. If is an odd prime, then

Proof. The proof is divided into four cases.

Case 1. .

By Theorem 1.4. (b) the prime can be written as

where . Then we have

From Theorem 2.1. we obtain . Since and by Theorem 1.1. the product of two quadratic residues is a quadratic residue modulo , it follows that

Case 2. .

From Theorem 1.4. (b), it follows that the congruence

has no solution. Then, for every , we obtain

Thus, we have . Since , and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 3. .

In the same way as in part (a), we have , and

From Theorem 2.1. we obtain . Since and by Theorem 1.1. the product of a quadratic residues with a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 4. .

In the same way as in part (b), for every , we have

Thus, . Now, since , and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo , it follows that

Theorem 2.4. If is an odd prime, then

Proof. The odd prime may take one of the forms

The proof is divided into four cases.

Case 1. .

By Theorem 2.1, the number is a quadratic residue modulo . Now, by Theorem 1.4. (c), the prime can be written as

where . Then we have

Since , and by Theorem 1.1. the product of two quadratic residues is a quadratic residue modulo , it follows that

Case 2. .

By Theorem 2.1, we have . From Theorem 1.4. (c), it follows that the congruence

has no solution. Then, for every , we obtain

Thus, we have . Since , and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 3. .

By Theorem 2.1, we have . From Theorem 1.4. (c), it follows that for every , , from which we obtain . Since , and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo , it follows that

Case 4. .

By Theorem 2.1, we have . From Theorem 1.4. (c), it follows that for every , , that is . Since , and the product of two quadratic non-residue is a quadratic residue modulo , it follows that

Example 2.5. Let be the prime .

The quadratic character of -1 modulo . We have

and

It follows that

The quadratic character of modulo . We have

and

It follows that

The quadratic character of modulo . We have

and

It follows that

The quadratic character of modulo . We have

and

It follows that

3. Quadratic Character of the Number and for Primes of the form and

In this section, we determine the quadratic character of the natural numbers and in the representations of primes of the forms

and

Theorem 3.1. Let be a prime with , and let such that . Then

(a) If , then .

(b) If , then the integers and have different quadratic characters modulo .

Proof. Since , we have

from which we obtain

Thus, the number .. is a quadratic residue modulo .

(a) . From Theorem 2.2. we have . Since the product of two quadratic residues is a quadratic residue modulo , it follows that .

It is clear that the integers and have different parity. Without loss of generality, suppose that is odd. Since and , we have

which shows that

Based on Theorem 1.5, we have

from which it follows that

Finally, since and , it follows that .

(b) . From Theorem 2.2. we have . Since and the product of a quadratic residue with a quadratic non-residue is a quadratic non-residue, it follows that ... Now, it is clear that the integers and have different quadratic characters modulo .

Example 3.2. For the prime , we are in the setting of Theorem 3.1 (a). We have

and also

For the prime , we are in the setting of Theorem 3.1 (b). We have

and also

and

Theorem 3.3. Let be a prime with and let such that . Then

(a) If , then .

(b) If , then and .

Proof. We first show that . Taking into account that, by Theorem 2.2, , in order to prove that is a quadratic residue, it suffices to prove that every odd prime divisor of is a quadratic residue modulo . Since

it follows that

Thus, is a quadratic residue modulo , so.

Now, since , we also have and by the Law of Quadratic Reciprocity, we obtain

Thus, the odd prime is a quadratic residue modulo . Finally, since the product of quadratic residues is again a quadratic residue, it follows that is a quadratic residue modulo .

By Theorem 2.2, we have . It follows that, there exists an integer such that . Since , we have

from which we obtain

This shows that . Since , it follows that

(a) If , then, is a quartic residue modulo . Hence, is a quadratic residue modulo . Thus, since and , it follows that .

(b) If , then, is a quartic non-residue modulo . Hence, is a quadratic non-residue modulo . Thus, from and , it follows that .

Note. Following Ireland and Rosen ⁷, the quartic residue character of integer modulo an odd prime can be characterized in terms of representations of by the quadratic form . More precisely, is a quartic residue modulo if and only if

for some integers , .

Thus, the condition

in Theorem 3.3. which shows that the number is a fourth power residue modulo , can be replaced by the condition stated above.

Let now be a prime with and let such that . As illustrated by the following example, the integers and may occur in any of the following four cases:

Example 3.4. For the prime we have

and also .

For the prime we have

and also and .

For the prime we have

and also and .

For the prime we have

and also .

We conclude by giving a theoretical description of the cases considered above for the integers and .

Let us first show that

From Theorems 2.1. and 2.2, we have , from which we obtain , that is . Let us denote for some integer . We have

and

Therefore, by Theorem 1.6, is a fourth-power residue modulo .

Now, let us denote by an integer such that

From the equality , we have:

from which we obtain or

If , then for the integers and , the cases and may occur, since the product of two quadratic residues is a quadratic residue, and the product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo .

If , then for the integers and , the cases and may occur, since product of a quadratic residue and a quadratic non-residue is a quadratic non-residue modulo .

References