On the non-linear diophantine equation \({\boldsymbol{379}}^{\boldsymbol{x}}\boldsymbol{+}{\boldsymbol{397}}^{\boldsymbol{y}}\boldsymbol{=}{\boldsymbol{z}}^{\boldsymbol{2}}\)

1. Introduction

Diophantine equations are those equations which are to be solved in integers. Diophantine equations are very important equations of theory of numbers and have many important applications in algebra, analytical geometry and trigonometry. These equations give us an idea to prove the existence of irrational numbers [1,2]. Acu [3] studied the diophantine equation \(2^x+5^y=z^2\) and proved that \(\left\{x=3,y=0,z=3\ \right\}\) and \(\left\{x=2,y=1,z=3\ \right\}\) are the solutions of this equation. Kumar et al., [4] considered the non-linear diophantine equations \({61}^x+{67}^y=z^2\) and \({67}^x+{73}^y=z^2\). They showed that these equations have no non-negative integer solution. Kumar et al. [5] studied the non-linear diophantine equations \({31}^x+{41}^y=z^2\) and \({61}^x+{71}^y=z^2\) and determined that these equations have no non-negative integer solution. Rabago [6] discussed the open problem given by B. Sroysang. He showed that the diophantine equation \(8^x+p^y=z^2,\) where \(x,y,z\) are positive integers has only three solutions namely \(\left\{x=1,y=1,z=5\ \right\},\) \(\left\{x=2,y=1,z=9\ \right\}\) and \(\left\{x=3,y=1,z=23\ \right\}\) for \(p=17\). The diophantine equations \(8^x+{19}^y=z^2\) and \(8^x+{13}^y=z^2\) were studied by Sroysang [7,8]. He proved that these equations have a unique non-negative integer solution namely \(\left\{x=1,y=0,z=3\ \right\}\). Sroysang [9] proved that the diophantine equation \({31}^x+{32}^y=z^2\) has no non-negative integer solution.

The main aim of this article is to discuss the existence of solution of non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers.

2. Preliminary

Lemma 1. The non-linear diophantine equation \({379}^x+1=z^2,\) where \(x,z\) are non-negative integers, has no solution in non-negative integers.

Proof. Since \(379\) is an odd prime, so \({379}^x\) is an odd number for all non-negative integer \(x\), which implies \({379}^x+1=z^2\) is an even number for all non-negative integer \(x\), so \(z\) is an even number. Hence

Now, \( 379=1\left(mod3\right)\), which implies \({379}^x=1\left(mod3\right),\) for all non-negative integer \(x\), so \({379}^x+1=2\left(mod3\right),\) for all non-negative integer \(x\). Hence Equation (2) contradicts Equation (1). Hence non-linear diophantine equation \({379}^x+1=z^2\) has no non-negative integer solution.

Lemma 2. The non-linear Diophantine equation \({397}^y+1=z^2,\) where \(y,z\) are for all non-negative integers, has no solution in non-negative integers.

Proof. Since \(397\) is an odd prime, so \({397}^y\) is an odd number for all non-negative integer \(y\), which implies \({397}^y+1=z^2\) is an even number for all non-negative integer \(y\), so \(z\) is an even number. Hence

Now, \(397=1\left(mod3\right)\), implies \({397}^y=1\left(mod3\right),\) for all non-negative integer \(y\), so \({397}^y+1=2\left(mod3\right),\) for all non-negative integer \(y\). Hence Equation (4) contradicts Equation (3). Hence non-linear Diophantine equation \({397}^y+1=z^2\) has no non-negative integer solution.

Theorem 1(Main theorem). The non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers, has no solution in non-negative integers.

Proof. There are following four cases;

• Case 1. If \(x=0\) then the non-linear diophantine equation \({379}^x+{397}^y=z^2\) becomes \(1+{397}^y=z^2\), which has no non-negative integer solution by Lemma 2.
• Case 2. If \(y=0\) then the non-linear diophantine equation \({379}^x+{397}^y=z^2\) becomes \({379}^x+1=z^2\), which has no non-negative integer solution by Lemma 1.
• Case 3. If \(x,y\) are positive integers, then \({379}^x,{397}^y\) are odd numbers, implies \( {379}^x+{397}^y=z^2\) is an even number, so \(z\) is an even number, Hence
  \begin{equation} \label{GrindEQ__5_} {z}^2=0\left(mod3\right) or z^2=1\left(mod3\right). \end{equation}

  (5)
  &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Now, \(379=1\left(mod3\right)\), implies \({379}^x=1\left(mod3\right)\) and \({397}^y=1\left(mod3\right)\), so \({379}^x+{397}^y=2\left(mod3\right)\). Hence
  \begin{equation} \label{GrindEQ__6_} z^2=2\left(mod3\right). \end{equation}

  (6)
  &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp Equation (6) contradicts Equation (5). Hence non-linear Diophantine equation \({379}^x+{397}^y=z^2\) has no non-negative integer solution.
• Case 4. If \(x,y=0\), then \({379}^x+{397}^y=1+1=2=z^2\), which is not possible because \(z\) is a non-negative integer. Hence diophantine equation \({379}^x+{397}^y=z^2\) has no non-negative integer solution.

3. Conclusion

In this article, authors successfully discussed the solution of non-linear diophantine equation \({379}^x+{397}^y=z^2\), where \(x,y,z\) are non-negative integers and determined that this non-linear equation has no non-negative integer solution.

Authorcontributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflict of Interests

”The authors declare no conflict of interest.”