The Diophantine Equation 8^x + p^y = z²

Abstract

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ±3(mod 8), then the equation 8x + p y = z 2 has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod 8), then the equation has only the solutions (p, x, y, z) = (2q − 1, (1/3)(q + 2), 2, 2q + 1), where q is an odd prime with q ≡ 1(mod 3); (iii) if p ≡ 1(mod 8) and p ≠ 17, then the equation has at most two positive integer solutions (x, y, z).

On the Diophantine equation 8^x + 113^y = z^2

“…Further, several Diophantine equations of different types have been studied by different workers [7,8,9,10,11,12,13]. Recently, Qi and Li [14] established that the Diophantine equation 8 x +p y = z 2 , x, y, z belong to natural number and p is an odd integer, with p ≡ 1(mod 8) and p = 17, have at most two positive integer solutions in (x, y, z) where p is an odd prime. Hence, it is a matter of further investigation to examine that, apart from p = 17, how many other such Diophantine equations are there which do not obey Qi and Li's [14] generalization.…”

“…Recently, Qi and Li [14] established that the Diophantine equation 8 x +p y = z 2 , x, y, z belong to natural number and p is an odd integer, with p ≡ 1(mod 8) and p = 17, have at most two positive integer solutions in (x, y, z) where p is an odd prime. Hence, it is a matter of further investigation to examine that, apart from p = 17, how many other such Diophantine equations are there which do not obey Qi and Li's [14] generalization. Although a number of other Diophantine equations has been solved by several other authors, yet it is imperative to search many more Diophantine equations violating Qi and Li's generalization [14], we have made an attempt to solve the new Diophantine equation containing p = 113, hitherto uninvestigated by any researcher to the best of our knowledge, and have found that it has three exact solutions in nonnegative integers (x, y, z).…”

“…Hence, it is a matter of further investigation to examine that, apart from p = 17, how many other such Diophantine equations are there which do not obey Qi and Li's [14] generalization. Although a number of other Diophantine equations has been solved by several other authors, yet it is imperative to search many more Diophantine equations violating Qi and Li's generalization [14], we have made an attempt to solve the new Diophantine equation containing p = 113, hitherto uninvestigated by any researcher to the best of our knowledge, and have found that it has three exact solutions in nonnegative integers (x, y, z). This problem constitutes second exception, apart from that of the Diophantine equation 8 x + 17 y = z 2 by Rabago [6] to the Qi and Li's generalizations regarding solutions of Diophantine equations of the type 8 x + p y = z 2 with p ≡ 1(mod 8) where p is a prime.…”