Abstract
Let p be an odd prime and \(\ell \ge 1\) an integer. In this paper, we prove that the class numbers of \(\mathbb {Q}(\sqrt{p^{12\ell +2}-4})\) and \(\mathbb {Q}(\sqrt{(4-p^{12\ell }+2)/3})\) are divisible by 3. Using Siegel’s theorem, we show that there are infinitely many such pairs of quadratic fields with class number divisible by 3.
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2010 Mathematics Subject Classification
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Acknowledgements
The authors would like to thank Azizul Hoque for suggesting the problem and also for a careful reading of this manuscript. The authors are also indebted to the anonymous for the thorough perusal of this paper.
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Kalita, H., Saikia, H.K. (2020). A Pair of Quadratic Fields with Class Number Divisible by 3. In: Chakraborty, K., Hoque, A., Pandey, P. (eds) Class Groups of Number Fields and Related Topics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1514-9_13
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