Abstract
In this article, authors discussed the existence of solution of non-linear diophantine equation \({379}^x+{397}^y=z^2,\) where \(x,y,z\) are non-negative integers. Results show that the considered non-linear diophantine equation has no non-negative integer solution.
Highlights
Diophantine equations are those equations which are to be solved in integers
Kumar et al, [4] considered the non-linear diophantine equations 61x + 67y = z2 and 67x + 73y = z2. They showed that these equations have no non-negative integer solution
He showed that the diophantine equation 8x + py = z2, where x, y, z are positive integers has only three solutions namely {x = 1, y = 1, z = 5 }, {x = 2, y = 1, z = 9 } and {x = 3, y = 1, z = 23 } for p = 17
Summary
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. Kumar et al, [4] considered the non-linear diophantine equations 61x + 67y = z2 and 67x + 73y = z2 They showed that these equations have no non-negative integer solution. Kumar et al [5] studied the non-linear diophantine equations 31x + 41y = z2 and 61x + 71y = z2 and determined that these equations have no non-negative integer solution. The diophantine equations 8x + 19y = z2 and 8x + 13y = z2 were studied by Sroysang [7,8] He proved that these equations have a unique non-negative integer solution namely {x = 1, y = 0, z = 3 }. The main aim of this article is to discuss the existence of solution of non-linear diophantine equation 379x + 397y = z2, where x, y, z are non-negative integers
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