Abstract
Bumby proved that the only positive solutions to the quartic Diophantine equation 3x 4 − 2y 2 = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field \({{\rm Q}(\sqrt{1-3a^{2}})}\) is not divisible by 2, the equation 3a 2 x 4 − By 2 = 1 has at most two solutions. However, both solutions occur in only one case, a = 1, b = 2, as solved by Bumby. The proof utilizes the law of quadratic reciprocity that seems very rare in solving Diophantine equations, and the solution will be also obtained effectively through the proof when it exists.
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He, D., Chen, J. & Wang, Y. On Diophantine equation 3a 2 x 4 − By 2 = 1. Annali di Matematica 189, 679–687 (2010). https://doi.org/10.1007/s10231-010-0131-8
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DOI: https://doi.org/10.1007/s10231-010-0131-8