On The Exponential Diophantine Equation (〖19〗^2m )+(6γ+1)^n=ρ^2

International Journal of Research and Innovation in Applied Science (IJRIAS) | Volume VI, Issue III, March 2021|ISSN 2454-6194

On The Exponential Diophantine Equation (〖19〗^2m )+(6γ+1)^n=ρ^2

Sudhanshu Aggarwal¹,Sanjay Kumar^2
1Assistant Professor, Department of Mathematics, National Post Graduate College, Barhalganj, Gorakhpur-273402, U.P., India
²Associate Professor, Department of Mathematics, M. S. College, Saharanpur-247001, U.P., India

Abstract: The family of Diophantine equations is divided into two categories (linear Diophantine equations and non-linear Diophantine equations). Diophantine equations are very useful for determining the solutions of many puzzle problems. In the present paper, authors studied the exponential Diophantine equation (〖19〗^2m )+(6γ+1)^n=ρ^2, where m,n,γ,ρ are whole numbers, for determining its solution in whole number. Results show that the exponential Diophantine equation (〖19〗^2m )+(6γ+1)^n=ρ^2, where m,n,γ,ρ are whole numbers, has no solution in whole number.

Keywords: Exponential Diophantine equation; Congruence; Modulo system; Numbers.

Mathematics Subject Classification: 11D61, 11D72, 11D45.

Introduction: Diophantine equations have many applications in the different field of mathematics such as coordinate geometry, cryptography, trigonometry and applied algebra. There is no generalizing method for solving all Diophantine equations. So, the problem of finding the solutions of Diophantine equations has very much attention by the scholars. Aggarwal et al. [1] discussed the Diophantine equation 〖223〗^x+〖241〗^y=z^2 for solution. Existence of solution of Diophantine equation 〖181〗^x+〖199〗^y=z^2 was given by Aggarwal et al. [2]. Bhatnagar and Aggarwal [3] proved that the exponential Diophantine equation 〖421〗^p+〖439〗^q=r^2 has no solution in whole number.
Gupta and Kumar [4] gave the solutions of exponential Diophantine equation n^x+〖(n+3m)〗^y=z^2k. Kumar et al. [5] studied exponential Diophantine equation 〖601〗^p+〖619〗^q=r^2 and proved that this equation has no solution in whole number. The non-linear Diophantine equations 〖61〗^x+〖67〗^y=z^2 and 〖67〗^x+〖73〗^y=z^2 are studied by Kumar et al. [6]. They determined that the equations

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