Abstract
It is well-known that the Pell equation \(t^2-Du^2=4\) has infinitely many integer solutions (t, u) for a given \(D>0\) with \(D\equiv 0,1\bmod {4}\) and \(D\ne l^2\) for any integer l. However, for a square free integer \(N\ne 1\), the equation \(t^2-Du^2=4N\) does not always have integer solutions, and verifying its solubility/insolubility is not easy. In the present paper, we propose the asymptotic formulas of the sum of the class numbers h(D) of the primitive indefinite binary quadratic forms over the discriminants \(D>0\) for which the Pell type equation \(t^2-Du^2=4N\) has an integer solution, to study the distributions of such D.
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The author would like to thank the anonymous referee for reading the previous draft of this paper carefully and giving helpful comments. He was supported by JST CREST no. JPMJCR14D6 and JSPS Grant-in-Aid for Scientific Research (C) no. 17K05181.
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Hashimoto, Y. Asymptotic behaviors of class number sums associated with Pell-type equations. Math. Z. 292, 641–654 (2019). https://doi.org/10.1007/s00209-018-2139-5
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DOI: https://doi.org/10.1007/s00209-018-2139-5