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On the diophantine equation (5 pn 2 - 1) x

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2020-03-01

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Honam Mathematical Soc

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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 (5pn(2) - 1)(x) + (p(p - 5)n(2) + 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 (35n(2) - 1)(x) + (14n(2) + 1)(y) = (7n)(z) has only the positive integer solution (x, y, z) = (1; 1; 2) where n +/- 3 (mod 5) or 5 vertical bar n. On the proofs, we use the properties of Jacobi symbol and Baker's method.

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Linear-forms, 2 logarithms, Conjecture, Exponential diophantine equation, Jacobi symbol, Baker's method, Science & technology, Physical sciences, Mathematics

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