On the Diophantine Equation (5^n )^x+(4^m p+1)〗^y=z^2

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

On the Diophantine Equation (5n )x+(4m p+1)y=z2

Vipawadee Moonchaisook1, Watakarn Moonchaisook2 and Khattiya Moonchaisook3
1Department of Mathematics Faculty of Science and Technology Surindra Rajabhat University, Surin 3200, Thailand.
2Computer Technology, Faculty of Agriculture and Technology, Rajamangala University of Technology Isan, Surin campus, Surin 32000,Thailand.
3Science and Mathematics, Faculty of Agriculture and Technology, Rajamangala University of Technology Isan, Surin campus, Surin 32000, Thailand.

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Abstract. In this paper, we proved that the Diophantine equation (5n )x+(4m p+1)y=z2 has no solution in non-negative integers x, y, z where p is an odd prime and m, n is a natural number.

Keywords: Diophantine equations, exponential equations, integer solution.

Ι.Introduction

Diophantine equation is one of the significant problems in elementary number theory and algebraic number theory. The Diophantine equation of the type a^x+b^y=z^2 has been studied by many authors for many years. In 2012, Sroysang [16] proved that the Diophantine equation 〖 3〗^x+5^y=z^2 has a unique non-negative integer solution where x, y and z are non-negative integers. The solution (x, y, z) is (1, 0, 2). In the same year, Sroysang [17] proved that the Diophantine equation 〖31〗^x+〖32〗^y=z^2 has no non-negative integer solution.
In 2017, Asthana, S., and Singh, M. M. [3] studies the Diophantine Equation 3^x+〖13〗^y=z^2 and proved that this has exactly four non-negative integer solutions for x, y and z. The solutions are (1, 0, 2), (1, 1, 4), (3, 2, 14) and (5, 1, 16) respectively. In 2018, Kumar et al. [10] studied the non-linear Diophantine equations 〖 61〗^x+〖67〗^y=z^2 and 〖 67〗^x+〖73〗^y=z^2 . They proved that these equations have no non-negative integer solution. Additionally, Kumar et al. [11] 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. In the same year, Burshtein N.[8] examined the solutions to the Diophantine Equation M^x+(M+6)^y=z^2 when M = 6N + 5 and M, M + 6 are primes. They proved that this equation has no solutions.