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http://dx.doi.org/10.5831/HMJ.2020.42.1.139

ON THE DIOPHANTINE EQUATION (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z  

Kizildere, Elif (Department of Mathematics, Bursa Uludag University)
Soydan, Gokhan (Department of Mathematics, Bursa Uludag University)
Publication Information
Honam Mathematical Journal / v.42, no.1, 2020 , pp. 139-150 More about this Journal
Abstract
Let p be a prime number with p > 3, p ≡ 3 (mod 4) and let n be a positive integer. In this paper, we prove that the Diophantine equation (5pn2 - 1)x + (p(p - 5)n2 + 1)y = (pn)z has only the positive integer solution (x, y, z) = (1, 1, 2) where pn ≡ ±1 (mod 5). As an another result, we show that the Diophantine equation (35n2 - 1)x + (14n2 + 1)y = (7n)z has only the positive integer solution (x, y, z) = (1, 1, 2) where n ≡ ±3 (mod 5) or 5 | n. On the proofs, we use the properties of Jacobi symbol and Baker's method.
Keywords
Exponential Diophantine equation; Jacobi symbol; Baker's method;
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