On the exponential diophantine equation (n-1)(x) + (n+2)(y) = n(z)
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Date
2020-03-30
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Ars Polona-Ruch
Abstract
Suppose that n is a positive integer. We show that the only positive integer solutions (n, x, y, z) of the exponential Diophantine equation
(n - 1)(x) + (n + 2)(y) = nz, n >= 2, xyz not equal 0,
are (3, 2, 1, 2), (3,1, 2, 3). The main tools in the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.
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Keywords
Exponential Diophantine equation, Primitive divisors of Lucas sequences, Jacobi symbol, Lower bounds for linear forms in two logarithms, Primitive divisors, Linear-forms, 2 Logarithms, Conjecture, Number, Lucas, Mathematics
Citation
Bai, H. vd. (2020). "On the exponential diophantine equation (n-1)(x) + (n+2)(y) = n(z)". Colloquium Mathematicum, 161(2), 239-249.