On the Diophantine Equation

Kishan, H.; Rani, Megha; Sarita, Sarita

doi:10.24297/jam.v11i9.815

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2022

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“…A number of researchers have studied the exponential Diophantine equations in the period 2010-2016 AD. Suvarnamani [29], Suvarnamani et al [30], Sroysang [27], Kishan et al [18], Bravo and Luca [5] etc. have done their extensive works on the different types of Diophantine equations 4 x + 7 y = z 2 , 4 x + 11 y = z 2 , 4 x + 13 y = z 2 , 4 x + 17 y = z 2 , A x + B y = C z , 3 x + 5 y = z 2 , 8 x + 19 y = z 2 , 31 x + 32 y = z 2 , 7 x + 8 y = z 2 , F n + F m = 2 a etc.…”

Section: Introductionmentioning

confidence: 99%

On Two Classes of Exponential Diophantine Equations

Borah

Dutta²

2022

Comm. Math. Appl.

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In this paper, we study on the exponential Diophantine equations: n x + 24 y = z 2 , for n ≡ 5 or 7 (mod 8). We show that 5 x + 24 y = z 2 has a unique positive integral solution (2, 1, 7). Further, we show that for k ∈ N, (8k + 5) x + 24 y = z 2 has a unique solution (0, 1, 5) in non-negative integers. We also show that for a perfect square 8m, the exponential Diophantine equation (8m − 1) x + 24 y = z 2 , m ∈ N has exactly two non-negative integral solutions (0, 1, 5) and (1, 0, 8m). Otherwise, it has a unique solution (0, 1, 5). Finally, we illustrate our results with some examples and non-examples.

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