On The Exponential Diophantine Equation (2^(2m+1)-1)+(6r+1)^n=z^2

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

On The Exponential Diophantine Equation (2^(2m+1)-1)+(6r+1)^n=z^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: 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. These equations help us for finding the integer solution of famous Pythagoras theorem and Pell’s equation. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. In the present paper, authors discussed the existence of the solution of exponential Diophantine equation (2^(2m+1)-1)+(6r+1)^n=z^2, where m,n,r,z are whole numbers.

Keywords: Positive integer; Diophantine equation; Solution; Congruence; Modulo system.

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

Introduction: 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. Diophantine equations are very helpful to prove the existence of irrational numbers [4, 6]. Acu [1] studied the Diophantine equation 2^x+5^y=z^2 and proved that {x=3,y=0,z=3 } and {x=2,y=1,z=3 } are the solutions of this equation. Kumar et al. [2] 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. [3] studied the non-linear Diophantine equations