On the diophantine equation 8a + 67ß = ?2

On the Nonexistence of Solutions to a Diophantine Equation Involving Prime Powers

“…Notably, recent research has seen a surge in interest in Diophantine equations of the form x r + y s = z 2 , where x, y are positive integers, and r, s , z are non-negative integers. Pioneering work by Sroysang [ [6], [7], [1], [2] ] has successfully solved this equation for various pairs (x,y), offering solutions for cases such as (3,5), (3,17), (3,17), (3,85), (3,45),(143,145) and others. Sroysang has also addressed the equation for the pairs (7,8) and (31,32) [ [7], [5]], along with posing a conjecture related to the equation p r + (p + 1) s = z 2 .Building upon this foundation, Chotchaisthit [3] made significant strides in 2013 by determining all solutions of the equation where p is the Mersenne prime, and specific forms of solutions were elucidated in [4] .…”

Solution of the Diophantine Equation

“…Aggarwal and Kumar [6] examined the exponential Diophantine equation (13 2m ) + (6r + 1) n = z 2 . Aggarwal and Upadhyaya [7] studied the Diophantine equation 8 α + 67 β = γ 2 and proved that this Diophantine equation has a unique solution in non-negative integers. Gupta et al [8] examined the Diophantine equation M 5 p + M 7 q = r 2 .…”

Solution of the Exponential Diophantine Equation 10^x + 400^y=z^2

“…Aggarwal and Upadhyaya [18] investigated the Diophantine equation 8 𝛼 + 67 𝛽 = 𝛾 2 and determined that it has a single nonnegative integer solution. Goel et al [19] used the arithmetic modular technique to investigate the Diophantine problem 𝑀 5 𝑝 + 𝑀 7 𝑞 = 𝑟 2 .…”