On The Exponential Diophantine Equation 〖M_3〗^p+〖M_5〗^q=r^2

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

On On The Exponential Diophantine Equation 〖M_3〗^p+〖M_5〗^q=r^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: Nowadays, researchers are very interested to determine the solution of different Diophantine equations because these equations have many applications in the field of coordinate geometry, trigonometry and applied algebra. These equations help us for finding the integer solution of famous Pythagoras theorem. Finding the solution of Diophantine equations have many challenges for scholars due to absence of generalize methods. In the present paper, authors discussed the exponential Diophantine equation 〖M_3〗^p+〖M_5〗^q=r^2, where p,q,r are whole numbers, M_3 and M_5 are Mersenne primes, for existence of its solution.

Keywords: Prime number; Diophantine equation; Solution; Mersenne primes, Lucas-Lehmer test.

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

Introduction: Diophantine equations are those equations of theory of numbers which are to be solved in integers. Diophantine equations are classified in two general 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. These 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 〖31〗^x+〖41〗^y=z^2 and 〖61〗^x+〖71〗^y=z^2. They determined that these equations have no non-negative integer solution. Rabago [5] discussed the open problem given by B. Sroysang. He determined that the Diophantine equation 8^x+p^y=z^2, 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. 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 solution which is given by {x=1,y=0,z=3 }. Sroysang [9] proved that the exponential Diophantine equation 〖31〗^x+〖32〗^y=z^2 has no positive integer solution. Aggarwal et al. [10] discussed the existence of solution of Diophantine equation 〖181〗^x+〖199〗^y=z^2. Aggarwal et al. [11] discussed the Diophantine equation 〖223〗^x+〖241〗^y=z^2 for solution.

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