Solution oSolution of Exponential Diophantine Equation n^x +43^y = z², where n≡2(mod 129) and n+1 is not a Perfect Square

Abstract

Nowadays, researchers are very interested in studying various Diophantine equations due to their importance in Cryptography, Chemistry, Knot Theory, Astronomy, Geometry, Trigonometry, Biology, Algebra, Electrical Engineering, Economics, and Astrology. The present paper is about the non-negative integer solution of the exponential Diophantine equation n^x + 43^y = Z², where x, y, z are non-negative integers,  n is a positive integer with  n ≡2(mod 129) and n+1 is not a perfect square. The authors use the famous Catalan conjecture for this purpose. Results of the present paper indicate that 2, 3, 0, and 3 are the only required values of and  respectively, that satisfy the exponential Diophantine equation n^x + 43^y = Z², where x, y, z are non-negative integers,  n is a positive integer with n ≡ 2(mod 129) and n+1 is not a perfect square. The present technique of this paper proposes a new approach to solving the Diophantine equations, which is the main scientific contribution of this study, and it is very beneficial, especially for researchers, scholars, academicians, and people interested in the same field.