Non-existence of Solutions of Diophantine Equations of the Form 𝒑 +

Copyright©2019 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Numerous researches have been devoted in finding the solutions ( 𝑥𝑥 , 𝑦 , 𝑧𝑧 ), in the set of non-negative integers, of Diophantine equations of type 𝑝𝑝 𝑥𝑥 + 𝑞𝑞 𝑦𝑦 = 𝑧𝑧 2 (1), where the values 𝑝𝑝 and 𝑞𝑞 are fixed. In this paper, we also deal with a more generalized form, that is, equations of type 𝑝𝑝 𝑥𝑥 + 𝑞𝑞 𝑦𝑦 = 𝑧𝑧 2𝑛𝑛 (2), where 𝑛𝑛 is a positive integer. We will present results that will guarantee the non-existence of solutions of such Diophantine equations in the set of positive integers. We will use the concepts of the Legendre symbol and Jacobi symbol, which were also used in the study of other types of Diophantine equations. Here, we assume that one of the exponents is odd. With these results, the problem of solving Diophantine equations of this type will become relatively easier as compared to the previous works of several authors. Moreover, we can extend the results by considering the Diophantine equations in the set of positive integers.

The goal of this paper is to present an easier way of showing that certain Diophantine equations of type + = 2 , where and are fixed positive integers, may fail to have solutions in the set ℕ of positive integers. This is done by using the concepts of Legendre symbol and the Jacobi symbol.

Preliminaries
The following definitions and well known results in Number Theory are essential in our study. Definition 2. 1. Let be an odd prime and be an integer such that gcd( , ) = 1 . If the congruence 2 ≡ ( ) has a solution in , then is said to be a quadratic residue of . Otherwise, is called a quadratic nonresidue of . Definition 2. 2. (Legendre Symbol) Let be an odd prime and be an integer with gcd( , ) = 1. Then the Legendre symbol � � is defined to be is the prime factorization of , where the 's are not necessarily distinct, then the Jacobi symbol � � is defined to be where the symbols � � are Legendre symbols.
The proofs of the above lemmas can be seen in [9].
From the Generalized Quadratic Reciprocity Law, if and are odd integers with , > 1 and gcd( , ) = 1 then So if is an odd integer that is not necessarily prime and gcd(3, ) = 1, we have Moreover, we also know that which tells us that the Jacobi symbol � 3 � coincides with the Legendre symbol. The same thing goes with the other Jacobi symbols including � 5 � , where ≠ 5 is an odd prime.

Main Results
We now present the results of the study. Examples are given to illustrate the results.
Proof. Suppose that � � = −1 and assume in contrary that the equation + = 2 has a solution ( , , ) ∈ ℕ 3 with being odd. Then taking the equation modulo , we get that 2 ≡ ( ) has a solution. This implies that � � = 1 by Lemma 2.4. We then have which is a contradiction. Hence + = 2 has no solution if is odd. Using similar arguments, one can show that the Diophantine equation + = 2 , for any ∈ ℕ, has no solutions in ℕ. The proof is also similar for the case where � � = −1 and is odd.
If 17 is replaced by any positive integer congruent to 2 modulo 3, the equation, still has no solution wherever is odd.
We can generalize Theorem 2.1 by considering and to be any positive integers such that gcd( , ) = 1. This is stated in the next theorem.
Theorem 3. 4. Let and be positive integers such that gcd( , ) = 1 . If = 2 0 1 1 . . . is a prime factorization of > 1 , then the Diophantine equation + = 2 where is odd, has no solutions if at least one of the following is satisfied: The result still holds if and are replaced by and , respectively.
Hence, no conclusion can be drawn about the non-existence of solutions. However, if we use Theorem 3.4, since 3 is a factor of 81, we get that This means that the equation has no solution when is odd.
Example 3. 6. In [3], Sroysang showed that the Diophantine equations 8 + 19 = 2 has no positive integer solution. We can verify the result if is odd, since Also, if is odd, it has no solution since 8 | but = 19 ≢ 1 ( 8). So far, we have been dealing with the case where and are relatively prime. Here is the extension for the case where gcd( , ) > 1.
Proof. First, assume that condition ( ) is true. Suppose in contrary, that the Diophantine equation + = 2 has a solution when is odd. By taking modulo on the equation, we see that 2 ≡ ( ) has a solution. Note that gcd( , ) = 1. Hence the Legendre symbol