The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z |
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| Authors: | Jianping Wang Tingting Wang Wenpeng Zhang |
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| Affiliation: | 1. College of Science, Chang’an University, Xi’an, Shaanxi, P.R. China
2. Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China
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| Abstract: | Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x 2 + (3n 2 + 1) y = (4n 2 + 1) z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n 3 + 3n, 1, 3). |
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