On the Exponential Diophantine Equation (7^2m )+(6^(r+1)+1)^n=ω^2

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International Journal of Research and Scientific Innovation (IJRSI) | Volume VIII, Issue IV, April 2021 | ISSN 2321–2705

On the Exponential Diophantine Equation (7^2m )+(6^(r+1)+1)^n=ω^2

 Sudhanshu Aggarwal1, Sanjay Kumar2
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

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Abstract: Diophantine equations are those equations of theory of numbers which are to be solved in integers. The class of Diophantine equations is classified in two categories, one is linear Diophantine equations and the other one is non-linear Diophantine equations. Both categories of these equations are very important in theory of numbers and have many important applications in solving the puzzle problems. In the present paper, author discussed the existence of the solution of exponential Diophantine equation (7^2m )+(6^(r+1)+1)^n=ω^2, where m,n,r,ω are whole numbers.

Keywords: Exponential Diophantine equation; Congruence; Modulo system; Numbers.
Mathematics Subject Classification: 11D61, 11D72, 11D45.

Introduction: Nowadays, scholars are very interested to determine the solution of different Diophantine equations because these equations have many applications in the field of coordinate geometry, cryptography, trigonometry and applied algebra. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. Aggarwal et al. [1] discussed the Diophantine equation 〖223〗^x+〖241〗^y=z^2 for solution. Aggarwal et al. [2] discussed the existence of solution of Diophantine equation 〖181〗^x+〖199〗^y=z^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. Kumar et al. [6] 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