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

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+(4^m p+1)^y=z²

Vipawadee Moonchaisook¹, Watakarn Moonchaisook² and Khattiya Moonchaisook^3
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.

Abstract. In this paper, we proved that the Diophantine equation (5ⁿ )^x+(4^m 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.

Read More